Finite Differences

The essence of the concept of finite differences is embodied in the standard definition of the derivative

where instead of passing to the limit as h approaches zero, the finite spacing to the next adjacent point, , is used so that we get an approximation.

The difference formula can also be derived from Taylor's formula,

which is more useful since it provides an error estimate (assuming sufficient smoothness)

An important aspect of this formula is that must lie between and so that the error is local to the interval enclosing the sampling points. It is generally true for finite difference formulas that the error is local to the stencil, or set of sample points. Typically, for convergence and other analysis, the error is expressed in asymptotic form:

This formula is most commonly referred to as the first order forward difference. The backward difference would use .

Taylor's formula can easily be used to derive higher order approximations. For example, subtracting

from

and solving for gives the second order centered difference formula for the first derivative,

If the Taylor formulas above are expanded out one order farther and added and then combined with the formula just above, it is not difficult to derive a centered formula for the second derivative.

Note that the while having a uniform step size h between points makes it convenient to write out the formulas, it is certainly not a requirement. For example, the approximation to the second derivative is in general

where h corresponds to the maximum local grid spacing. Note that the asymptotic order of the three-point formula has dropped to first order; that it was second order on a uniform grid is due to fortuitous cancellations.

In general, formulas for any given derivative with asymptotic error of any chosen order can be derived from the Taylor formulas as long as a sufficient number of sample points are used. However, this method becomes cumbersome and inefficient beyond the simple examples shown above. An alternate formulation is based on polynomial interpolation: since the Taylor formulas are exact (no error term) for polynomials of sufficiently low order, so are the finite difference formulas. It is not difficult to show that the finite difference formulas are equivalent to the derivatives of interpolating polynomials. For example, a simple way of deriving the formula shown above for the second derivative is to interpolate a quadratic and find its second derivative (which is essentially just the leading coefficient).

This finds the three point finite difference formula for the second derivative by differentiating the polynomial interpolating the three points , , and

In this form of the formula, it is easy to see that it is effectively a difference of the forward and backward first order derivative approximations. Sometimes it is advantageous to use finite differences in this way, particularly for terms with coefficients inside of derivatives, such as (, which commonly appear in PDEs.

Another property made apparent by considering interpolation formulas is that the point at which you get the derivative approximation need not be on the grid. A common use of this is with staggered grids where the derivative may be wanted at the midpoints between grid points.

This generates a fourth order approximation for the first derivative on a uniform staggered grid, , where the main grid points are at +h k/2, for odd k.

The fourth order error coefficient for this formula is versus for the standard fourth order formula derived below. Much of the reduced error can be attributed to the reduced stencil size.

This generates a fourth order approximation for the first derivative at a point on a uniform grid.

In general, a finite difference formula using n points will be exact for functions that are polynomials of degree n - 1 and have asymptotic order at least n - m. On uniform grids, you can expect higher asymptotic order, especially for centered differences.

Using efficient polynomial interpolation techniques is a reasonable way to generate coefficients, but B. Fornberg has developed a fast algorithm for finite difference weight generation [F92], [F98], which is substantially faster.

In [F98], he presents a one-line Mathematica formula for explicit finite differences.

This is the simple formula of Fornberg for generating weights on a uniform grid. Here it has been modified slightly by making it a function definition.

Here m is the order of the derivative, n is the number of grid intervals enclosed in the stencil, and s is the number of grid intervals between the point at which the derivative is approximated and the leftmost edge of the stencil. There is no requirement that s be an integer; noninteger values simply lead to staggered grid approximations. Setting s to be n/2 always generates a centered formula.

This uses the Fornberg formula to generate the weights for a staggered fourth order approximation to the first derivative. This is the same one computed above with InterpolatingPolynomial.

A table of some commonly used finite difference formulas is given below for reference.

Finite difference formulas on uniform grids for the first derivative.

Finite difference formulas on uniform grids for the second derivative.

One thing to notice from this table is that the farther the formulas get from centered, the larger the error term coefficient, sometimes by factors of hundreds. For this reason, sometimes where one-sided derivative formulas are required (such as at boundaries), formulas of higher order are used to offset the extra error.

NDSolve`FiniteDifferenceDerivative

Fornberg [F92], [F98] also gives an algorithm that, though not quite so elegant and simple, is more general and, in particular, is applicable to nonuniform grids. It is not difficult to program in Mathematica, but to make it as efficient as possible, a new kernel function has been provided as a simpler interface (along with some additional features).

Finding finite difference approximations to derivatives.

This defines a uniform grid with points spaced apart by a symbolic distance h.

This gives the first derivative formulas on the grid for a symbolic function f.

The derivatives at the endpoints are computed using one-sided formulas. The formulas shown in the example above are fourth order accurate, which is the default. In general, when you use symbolic grid and/or data, you will get symbolic formulas. This is often useful for doing analysis on the methods; however, for actual numerical grids, it is usually faster and more accurate to give the numerical grid to NDSolve`FiniteDifferenceDerivative rather than using the symbolic formulas.

This defines a randomly spaced grid between 0 and 2 π.

This approximates the derivative of the sine function at each point on the grid.

This shows the error in the approximations.

In multiple dimensions, NDSolve`FiniteDifferenceDerivative works on tensor product grids, and you only need to specify the grid points for each dimension.

This defines grids xgrid and ygrid for the x and y direction, gives an approximation for the mixed xy partial derivative of the Gaussian on the tensor product of xgrid and ygrid, and makes a surface plot of the error.

Note that the values need to be given in a matrix corresponding to the outer product of the grid coordinates.

NDSolve`FiniteDifferenceDerivative does not compute weights for sums of derivatives. This means that for common operators like the Laplacian, you need to combine two approximations.

This makes a function that approximates the Laplacian operator on a tensor product grid.

This uses the function to approximate the Laplacian for the same grid and Gaussian function used in the previous example.

Options for NDSolve`FiniteDifferenceDerivative.

This approximates the derivatives for the sine function on the random grid defined above, assuming that the function repeats periodically.

When using PeriodicInterpolationTrue, you can omit the last point in the values since it should always be the same as the first. This feature is useful when solving a PDE with periodic boundary conditions.

This generates second order finite difference formulas for the first derivative of a symbolic function.

Fourth order differences typically provide a good balance between truncation (approximation) error and roundoff error for machine precision. However, there are some applications where fourth order differences produce excessive oscillation (Gibb's phenomena), so second order differences are better. Also, for high precision, higher order differences may be appropriate. Even values of DifferenceOrder use centered formulas, which typically have smaller error coefficients than non-centered formulas, so even values are recommended when appropriate.

NDSolve`FiniteDifferenceDerivativeFunction

When computing the solution to a PDE, it is common to repeatedly apply the same finite difference approximation to different values on the same grid. A significant savings can be made by storing the necessary weight computations and applying them to the changing data. When you omit the (third) argument with function values in NDSolve`FiniteDifferenceDerivative, the result will be an NDSolve`FiniteDifferenceDerivativeFunction, which is a data object that stores the weight computations in a efficient form for future repeated use.

Computing finite difference weights for repeated use.

This defines a uniform grid with 25 points on the unit interval and evaluates the sine function with one period on the grid.

This defines an NDSolve`FiniteDifferenceDerivativeFunction, which can be repeatedly applied to different values on the grid to approximate the second derivative.

Note that the standard output form is abbreviated and only shows the derivative operators that are approximated.

This computes the approximation to the second derivative of the sine function.

This function is only applicable for values defined on the particular grid used to construct it. If your problem requires changing the grid, you will need to use NDSolve`FiniteDifferenceDerivative to generate weights each time the grid changes. However, when you can use NDSolve`FiniteDifferenceDerivativeFunction objects, evaluation will be substantially faster.

This compares timings for computing the Laplacian with the function just defined and with the definition of the previous section. A loop is used to repeat the calculation in each case because it is too fast for the differences to show up with Timing.

An NDSolve`FiniteDifferenceDerivativeFunction can be used repeatedly in many situations. As a simple example, consider a collocation method for solving the boundary value problem

on the unit interval. (This simple method is not necessarily the best way to solve this particular problem, but it is useful as an example.)

This defines a function that will have all components zero at an approximate solution of the boundary value problem. Using the intermediate vector v and setting its endpoints (parts {1,-1}) to 0 is a fast and simple trick to enforce the boundary conditions. Evaluation is prevented except for numbers λ because this would not work otherwise. (Also, because of the Listability of Times, Sin[2 Pi grid] u would thread componentwise.)

This uses FindRoot to find an approximate eigenfunction using the constant coefficient case for a starting value and shows a plot of the eigenfunction.

The FindRoot::symdv message does not mean that FindRoot is unable to properly solve the problem. However, giving consideration to its import can lead to substantial improvements in the algorithm. These will be explored in the section below on differentiation matrices.

The setup for this problem is so simple, it is easy to compare various alternatives. For example, to compare the solution above, which used the default fourth order differences, to the usual use of second order differences, all that needs to be changed is the DifferenceOrder.

This solves the boundary value problem using second order differences and shows a plot of the difference between it and the fourth order solution.

One way to determine which is the better solution is to study the convergence as the grid is refined. This will be considered to some extent in the section on differentiation matrices below.

While the most vital information about an NDSolve`FiniteDifferenceDerivativeFunction object, the derivative order, is displayed in its output form, sometimes it is useful to extract this and other information from an NDSolve`FiniteDifferenceDerivativeFunction, say for use in a program. The structure of the way the data is stored may change between versions of Mathematica, so extracting the information by using parts of the expression is not recommended. A better alternative is to use any of the several method functions provided for this purpose.

Let FDDF represent an NDSolve`FiniteDifferenceDerivativeFunction[data] object.

Method functions for exacting information from an NDSolve`FiniteDifferenceDerivativeFunction[data] object.

Any of the method functions that return a list with an element for each of the dimensions can be used with an integer argument dim, which will return only the value for that particular dimension such that FDDF@method[dim] = (FDDF@method[])[[dim]].

The examples below show how you might use some of these methods.

Here is an NDSolve`FiniteDifferenceDerivativeFunction object created with random grids having between 10 and 16 points in each dimension.

This shows the dimensions of the outer product grid.

Note that the rank of the grid point tensor is one more than the dimensionality of the tensor product. This is because the 3 coordinates defining each point are in a list themselves. If you have a function that depends on the grid variables, you can use Apply[f, fddf@Grid[], {n}] where n = Length[fddf@DerivativeOrder[]] is the dimensionality of the space in which you are approximating the derivative.

This defines a Gaussian function of 3 variables and applies it to the grid on which the NDSolve`FiniteDifferenceDerivativeFunction is defined.

This shows a 3-dimensional plot of the grid points colored according to the scaled value of the derivative.

For a moderate sized tensor product grid like the example above, using Apply is reasonably fast. However, as the grid size gets larger, this approach may not be the fastest because Apply can only be used in limited ways with the Mathematica compiler and hence, with packed arrays. If you can define your function so you can use Map instead of Apply, you may be able to use a CompiledFunction since Map has greater applicability within the Mathematica compiler than does Apply.

This defines a CompiledFunction which using Map to get the values on the grid. (If the first grid dimension is greater than the SystemOption "MapCompileLength", then you do not need to construct the CompiledFunction since the compilation is done automatically when grid is a packed array.)

An even better approach, when possible, is to take advantage of listability when your function consists of operations and functions which have the Listable attribute. The trick is to separate the x, y, and z values at each of the points on the tensor product grid. The fastest way to do this is using Transpose[fddf@Grid[], RotateLeft[Range[n + 1]]], where n = Length[fddf@DerivativeOrder[]] is the dimensionality of the space in which you are approximating the derivative. This will return a list of length n, which has the values on the grid for each of the component dimensions separately. With the Listable attribute, functions applied to this will thread over the grid.

This defines a function that takes advantage of the fact that Exp has the Listable attribute to find the values on the grid.

This compares timings for the three methods. The commands are repeated several times to get more accurate timings.

The example timings show that using the CompiledFunction is typically much faster than using Apply and taking advantage of listability is a little faster yet.

Pseudospectral Derivatives

The maximum value the difference order can take on is determined by the number of points in the grid. If you exceed this, a warning message will be given and the order reduced automatically.

This uses maximal order to approximate the first derivative of the sine function on a random grid.

Using a limiting order is commonly referred to as a pseudospectral derivative. A common problem with these is that artificial oscillations (Runge's phenomena) can be extreme. However, there are two instances where this is not the case: a uniform grid with periodic repetition and a grid with points at the zeros of the Chebyshev polynomials, , or Chebyshev-Gauss-Lobatto points [F96], [QV94]. The computation in both of these cases can be done using a fast Fourier transform, which is efficient and minimizes roundoff error.

Settings for the DifferenceOrder option.

This gives a pseudospectral approximation for the second derivative of the sine function on a uniform grid.

This computes the error at each point. The approximation is accurate to roundoff because the effective basis for the pseudospectral derivative on a uniform grid for a periodic function are the trigonometric functions.

The Chebyshev-Gauss-Lobatto points are the zeros of . Using the property , these can be shown to be at .

This defines a simple function that generates a grid of n points with leftmost point at x0 and interval length L having the spacing of the Chebyshev-Gauss-Lobatto points

This computes the pseudospectral derivative for a Gaussian function.

This shows a plot of the approximation and the exact values.

This shows a plot of the derivative computed using a uniform grid with the same number of points with maximal difference order.

Even though the approximation is somewhat better in the center (because the points are more closely spaced there in the uniform grid), the plot clearly shows the disastrous oscillation typical of overly high order finite difference approximations. Using the Chebyshev-Gauss-Lobatto spacing has minimized this.

This shows a plot of the pseudospectral derivative approximation computed using a uniform grid with periodic repetition.

With the assumption of periodicity, the approximation is significantly improved. The accuracy of the periodic pseudospectral approximations is sufficiently high to justify, in some cases, using a larger computational domain to simulate periodicity, say for a pulse like the above example. Despite the great accuracy of these approximations, they are not without pitfalls: one of the worst is probably aliasing error, whereby an oscillatory function component with too great a frequency can be misapproximated or disappear entirely.

Accuracy and Convergence of Finite Difference Approximations

When using finite differences, it is important to keep in mind that the truncation error, or the asymptotic approximation error induced by cutting off the Taylor series approximation, is not the only source of error. There are two other sources of error in applying finite difference formulas; condition error and roundoff error [GMW81]. Roundoff error comes from roundoff in the arithmetic computations required. Condition error comes from magnification of any errors in the function values, typically from the division by a power of the step size, and so grows with decreasing step size. This means that in practice, even though the truncation error approaches zero as h does, the actual error will start growing beyond some point. The figures below demonstrate the typical behavior as h becomes small for a smooth function.

A logarithmic plot of the maximum error for approximating the first derivative of the Gaussian at points on a grid covering the interval [0,1] as a function of the number of grid points, n, using machine precision. Finite differences of order 2,4,6, and 8 on a uniform grid are shown in red, green, blue, and magenta respectively. Pseudospectral derivatives with uniform (periodic) and Chebyshev spacing are shown in black and grey respectively.

A logarithmic plot of the truncation error (dotted) and the condition and roundoff error (solid line) for approximating the first derivative of the Gaussian at points on a grid covering the interval [0,1] as a function of the number of grid points, n. Finite differences of order 2, 4, 6, and 8 on a uniform grid are shown in red, green, blue, and magenta, respectively. Pseudospectral derivatives with uniform (periodic) and Chebyshev spacing are shown in black and grey, respectively. The truncation error was computed by computing the approximations with very high precision. The roundoff and condition error was estimated by subtracting the machine precision approximation from the high precision approximation. The roundoff and condition error tends to increase linearly (because of the 1/h factor common to finite difference formulas for the first derivative) and tends to be a little bit higher for higher order derivatives. The pseudospectral derivatives show more variations because the error of the FFT computations vary with length. Note that the truncation error for the uniform (periodic) pseudospectral derivative does not decrease below about . This is because, mathematically, the Gaussian is not a periodic function; this error in essence gives the deviation from periodicity.

A semilogarithmic plot of the error for approximating the first derivative of the Gaussian as a function of x at points on a 45 point grid covering the interval [0,1]. Finite differences of order 2, 4, 6, and 8 on a uniform grid are shown in red, green, blue, and magenta, respectively. Pseudospectral derivatives with uniform (periodic) and Chebyshev spacing are shown in black and grey, respectively. All but the pseudospectral derivative with Chebyshev spacing were computed using uniform spacing 1/45. It is apparent that the error for the pseudospectral derivatives is not so localized; not surprising since the approximation at any point is based on the values over the whole grid. The error for the finite difference approximations are localized and the magnitude of the errors follows the size of the Gaussian (which is parabolic on a semilogarithmic plot).

From the second plot, it is apparent that there is a size for which the best possible derivative approximation is found; for larger h, the truncation error dominates, and for smaller h, the condition and roundoff error dominate. The optimal h tends to give better approximations for higher order differences. This is not typically an issue for spatial discretization of PDEs because computing to that level of accuracy would be prohibitively expensive. However, this error balance is a vitally important issue when using low order differences to approximate, for example, Jacobian matrices. To avoid extra function evaluations, first order forward differences are usually used, and the error balance is proportional to the square root of unit roundoff, so picking a good value of h is important [GMW81].

The plots above showed the situation typical for smooth functions where there were no real boundary effects. If the parameter in the Gaussian is changed so the function is flatter, boundary effects begin to appear.

A semilogarithmic plot of the error for approximating the first derivative of the Gaussian as a function of x at points on a 45 point grid covering the interval [0,1]. Finite differences of order 2, 4, 6, and 8 on a uniform grid are shown in red, green, blue, and magenta, respectively. Pseudospectral derivatives with uniform (nonperiodic) and Chebyshev spacing are shown in black and grey, respectively. All but the pseudospectral derivative with Chebyshev spacing were computed using uniform spacing 1/45. The error for the finite difference approximations are localized, and the magnitude of the errors follows the magnitude of the first derivative of the Gaussian. The error near the boundary for the uniform spacing pseudospectral (order 45 polynomial) approximation becomes enormous; as h decreases, this is not bounded. On the other hand, the error for the Chebyshev spacing pseudospectral is more uniform and overall quite small.

From what has so far been shown, it would appear that the higher the order of the approximation, the better. However, there are two additional issues to consider. The higher order approximations lead to more expensive function evaluations, and if implicit iteration is needed (as for a stiff problem), then not only is computing the Jacobian more expensive, but the eigenvalues of the matrix also tend to be larger, leading to more stiffness and more difficultly for iterative solvers. This is at an extreme for pseudospectral methods, where the Jacobian has essentially no nonzero entries [F96]. Of course, these problems are a trade-off for smaller system (and hence matrix) size.

The other issue is associated with discontinuities. Typically, the higher order the polynomial approximation, the worse the approximation. To make matters even worse, for a true discontinuity, the errors magnify as the grid spacing is reduced.

A plot of approximations for the first derivative of the discontinuous unit step function as a function of x at points on a 128 point grid covering the interval [0,1]. Finite differences of order 2, 4, 6, and 8 on a uniform grid are shown in red, green, blue, and magenta, respectively. Pseudospectral derivatives with uniform (periodic) and Chebyshev spacing are shown in black and grey, respectively. All but the pseudospectral derivative with Chebyshev spacing were computed using uniform spacing 1/128. All show oscillatory behavior, but it is apparent that the Chebyshev pseudospectral derivative does better in this regard.

There are numerous alternatives that are used around known discontinuities, such as front tracking. First order forward differences minimize oscillation, but introduce artificial viscosity terms. One good alternative are the so-called essentially nonoscillatory (ENO) schemes, which have full order away from discontinuities but introduce limits near discontinuities that limit the approximation order and the oscillatory behavior. At this time, ENO schemes are not implemented in NDSolve.

In summary, choosing an appropriate difference order depends greatly on the problem structure. The default of 4 was chosen to be generally reasonable for a wide variety of PDEs, but you may want to try other settings for a particular problem to get better results.

Differentiation Matrices

Since differentiation, and naturally finite difference approximation, is a linear operation, an alternative way of expressing the action of a FiniteDifferenceDerivativeFunction is with a matrix. A matrix that represents an approximation to the differential operator is referred to as a differentiation matrix [F96]. While differentiation matrices may not always be the optimal way of applying finite difference approximations (particularly in cases where an FFT can be used to reduce complexity and error), they are invaluable as aids for analysis and, sometimes, for use in the linear solvers often needed to solve PDEs.

Forming a differentiation matrix.

This creates a FiniteDifferenceDerivativeFunction object.

This makes a matrix representing the underlying linear operator.

The matrix is given in a sparse form because, in general, differentiation matrices have relatively few nonzero entries.

This converts to a normal dense matrix and displays it using MatrixForm.

This shows that all three of the representations are roughly equivalent in terms of their action on data.

As mentioned previously, the matrix form is useful for analysis. For example, it can be used in a direct solver or to find the eigenvalues that could, for example, be used for linear stability analysis.

This computes the eigenvalues of the differentiation matrix.

For pseudospectral derivatives, which can be computed using fast Fourier transforms, it may be faster to use the differentiation matrix for small size, but ultimately, on a larger grid, the better complexity and numerical properties of the FFT make this the much better choice.

For multidimensional derivatives, the matrix is formed so that it is operating on the flattened data. It is easiest to understand this through an example.

This evaluates a Gaussian function on the grid that is the outer product of grids in the x and y direction.

This defines an NDSolve`FiniteDifferenceDerivativeFunction which computes the mixed x-y partial of the function using fourth order differences.

This computes the associated differentiation matrix.

Note that the differentiation matrix is a 1353x1353 matrix. The number 1353 is the total number of points on the tensor product grid, that, of course, is the product of the number of points on the x and y grids. The differentiation matrix operates on a vector of data which comes from flattening data on the tensor product grid. The matrix is also very sparse; only about a half of a percent of the entries are nonzero. This is easily seen with a plot of the positions with nonzero values.

Load the MatrixManipulation package and show a plot of the positions with nonzero values for the differentiation matrix.

This compares the computation of the mixed x-y partial with the two methods.

Using the differentiation matrix results in slightly different values for machine numbers because the order of operations is different which, in turn, leads to different roundoff errors.

This compares timings for flattening the data and computing the mixed x-y partial with the two methods. A loop is used to repeat the calculation in each case because it is too fast for the differences to show up with Timing. It is also important to make sure that the data is a packed array first, so that the timing comparison is based on the actual operations and not on data manipulation.

While the cost of flattening the data is quite small, the difference between using the NDSolve`FiniteDifferenceDerivativeFunction and the differentiation matrix is quite significant. The reason is that while using the differentiation matrix takes advantage of the sparseness of the matrix in a general way, the NDSolve`FiniteDifferenceDerivativeFunction actually uses the particular sparse structure of the tensor product to gain an advantage. The actual number of operations done is the same; the difference is in better memory access, which may vary from processor to processor and will typically get larger as the problem size increases. Reordering variables (as might be done for a sparse linear solver anyway) can help to reduce, if not eliminate, this difference.

Even though the differentiation matrix in multiple dimensions is typically slower, there are some cases where using it can speed up an overall calculation. For example, the computation of the Laplacian operator can be put into a single matrix.

This makes a function that approximates the Laplacian operator on a the tensor product grid.

This computes the differentiation matrices associated with the derivatives in the x and y direction.

This adds the two sparse matrices together resulting in a single matrix for the Laplacian operator.

This shows a plot of the positions with nonzero values for the differentiation matrix.

This compares the values and timings for the two different ways of approximating the Laplacian.

The two timings are comparable on this processor for the chosen grid size. Typically, as problem size increases, the matrix approach will be increasingly less efficient unless some variable reordering is used.

NDSolve::eerr: "Warning: Scaled local spatial error estimate of \!\(498.01499096622746`\) at \!\(t\) = \!\(2.`\) in the direction of independent variable \!\(x\) is much greater than prescribed error tolerance. Grid spacing with \!\(19\) points may be too large to achieve the desired accuracy or precision. A singularity may have formed or you may want to specify a smaller grid spacing using the MaxStepSize or MinPoints options."

Overview

When NDSolve solves a PDE, unless you have specified the spatial grid for it to use, by giving it explicitly or by giving a equal values for the MinPoints and MaxPoints options, NDSolve needs to make a spatial error estimate.

Ideally, the spatial error estimates would be monitored over time and the spatial mesh updated according to the evolution of the solution. The problem of grid adaptivity is difficult enough for a specific type of PDE and certainly has not been solved in any general way. Furthermore, techniques such as local refinement can be problematic with the method of lines since changing the number of mesh points requires a complete restart of the ODE methods. There are moving mesh techniques that appear promising for this approach, but at this point, NDSolve uses a static grid. The grid to use is determined by an a-priori error estimate based on the initial condition. An a-posteriori check is done at the end of the temporal interval for reasonable consistency and a warning message is given if that fails. This can, of course, be fooled, but in practice it provides a reasonable compromise. The most common cause of failure is when initial conditions have little variation, so the estimates are essentially meaningless. In this case, you may need to choose some appropriate grid settings yourself.

A-priori Error Estimates

When NDSolve solves a PDE using the method of lines, a decision has to be made on an appropriate spatial grid. NDSolve does this using an error estimate based on the initial condition (thus, a-priori).

It is easiest to show how this works in the context of an example. For illustrative purposes, consider the sine-Gordon equation in 1-dimension with periodic boundary conditions.

This solves the sine-Gordon equation with a Gaussian initial condition.

This gives the number of spatial and temporal points used, respectively.

The temporal points are chosen adaptively by the ODE method based on local error control. NDSolve used 49 (50 including the periodic endpoint) spatial points. This choice will be illustrated through the steps that follow.

In the equation processing phase of NDSolve, one of the first things that happen is that equations with second (or higher) order temporal derivatives are replaced with systems with only first order temporal derivatives.

This is a first order system equivalent to the sine-Gordon equation above.

The next stage is to solve for the temporal derivatives.

This is the solution for the temporal derivatives, with the right hand side of the equations in normal (ODE) form.

Now the problem is to choose a uniform grid that will approximate the derivative to within local error tolerances as specified by AccuracyGoal and PrecisionGoal. For this illustration, use the default DifferenceOrder (4) and the default AccuracyGoal and PrecisionGoal (both 4 for PDEs). The methods used to integrate the system of ODEs that results from discretization base their own error estimates on the assumption of sufficiently accurate function values. The estimates here have the goal of finding a spatial grid for which (at least with the initial condition) the spatial error is somewhat balanced with the local temporal error.

This sets variables to reflect the default settings for DifferenceOrder, AccuracyGoal, and PrecisionGoal.

The error estimate is based on Richardson extrapolation. If you know that the error is and you have two approximations and at different values, and of h, then you can, in effect, extrapolate to the limit h->0 to get an error estimate

so the error in is estimated to be

Here and are vectors of different length and is a function, so we need to choose an appropriate norm. If we choose = 2 , then we can simply use a scaled norm on the components common to both vectors, which is all of and every other point of .This is a good choice because it does not require any interpolation between grids.

For a given interval on which we want to set up a uniform grid, we can define a function , where is the length of the interval such that the grid is , , ..., }, where .

This defines functions that return the step size and a list of grid points as a function of for the sine-Gordon equation.

For a given grid, the equation can be discretized using finite differences. This is easily done using NDSolve`FiniteDifferenceDerivative.

This defines a symbolic discretization of the right hand side of the sine-Gordon equation as a function of a grid. It returns a function of u and v, which gives the approximate values for and in a list. (Note that in principle this works for any grid, uniform or not, though in the following, only uniform grids will be used.)

For a given step size and grid, we can also discretize the initial conditions for u and v.

This defines a function that discretizes the initial conditions for u and v. The last grid point is dropped because, by periodic continuation, it is considered the same as the first.

The quantity of interest is the approximation of the right hand side for a particular value of with this initial condition.

This defines a function that returns a vector consisting of the approximation of the right hand side of the equation for the initial condition for a given step size and grid. The vector is flattened to make subsequent analysis of it simpler.

Starting with a particular value of , we can obtain the error estimate by generating the right hand side for and points.

This gives an example of the right hand side approximation vector for a grid with 10 points.

This gives an example of the right hand side approximation vector for a grid with 20 points.

As mentioned above, every other point on the grid with points lies on the grid with points. Thus, for simplicity, we can use a norm that only compares point common to both grids. Because the goal is to ultimately satisfy absolute and relative tolerance criteria, it is appropriate to use a scaled norm. In addition to taking into account the size of the right hand side for the scaling, it is also important to include the size of the corresponding components of u and v on the grid since error in the right hand side is ultimately included in u and v.

This defines a norm function for the difference in the approximation of the right hand side.

This applies the norm function to the two approximations found above.

To get the error estimate form the distance, according to the Richardson extrapolation formula (3), this simply needs to be divided by .

This computes the error estimate for n == 10. Since this is based on a scaled norm, the tolerance criteria are satisfied if the result is less than 1.

This makes a function that combines the above functions to give an error estimate as a function of n.

The goal is to find the minimum value of n, such that the error estimate is less than or equal to 1 (since it is based on a scaled norm). In principle, it would be possible to use a root finding algorithm on this, but since n can only be an integer, this would be overkill and adjustments would have to be made to the stopping conditions. An easier solution is simply to use the simple Richardson extrapolation formula to predict what value of n would be appropriate and repeat the prediction process until the appropriate n is found.

The condition to satisfy is

and we have estimated that

so we can project that

or in terms of n, which is proportional to the reciprocal of h,

This computes the predicted optimal value of n based on the error estimate for n == 10 computed above.

This computes the error estimate for the new value of n.

Often the case that a prediction based on a very coarse grid will be inadequate. A coarse grid may completely miss some solution features that contribute to the error on a finer grid. Also, the error estimate is based on an asymptotic formula, so for coarse spacings, the estimate itself may not be very good, even when all the solution features are resolved to some extent.

In practice, it can be fairly expensive to compute these error estimates. Also, it is not necessary to find the very optimal n, but one that satisfies the error estimate. Remember, everything can change as the PDE evolves, so it is simply not worth a lot of extra effort to find an optimal spacing for just the initial time. A simple solution is to include an extra factor greater than 1 in the prediction formula for the new n. Even with an extra factor, it may still take a few iterations to get to an acceptable value. It does, however, typically make the process faster.

This defines a function that gives a predicted value for the number of grid points, which should satisfy the error estimate.

This iterates the predictions until a value is found that satisfies the error estimate.

It is important to note that this process must have a limiting value since it may not be possible to satisfy the error tolerances, for example, with a discontinuous initial function. In NDSolve, the MaxSteps option provides the limit; for spatial discretization, this defaults to a total of 10000 across all spatial dimensions.

Pseudospectral derivatives cannot use this error estimate since they have an exponential rather than a polynomial convergence. An estimate can be made based on the formula used above in the limit p->Infinity. What this amounts to is considering the result on the finer grid to be exact and basing the error estimate on the difference since (1 - ) approaches 1. A better approach is to use the fact that on a given grid with n points, the pseudospectral method is . When comparing for two grids, it is appropriate to use the smaller n for p. This provides an imperfect, but adequate estimate for the purpose of determining grid size.

This modifies the error estimation function so that it will work with pseudospectral derivatives.

The prediction formula can be modified to use n instead of p in a similar way.

This modifies the function predicting an appropriate value of n to work with pseudospectral derivatives. This formulation does not try to pick an efficient FFT length.

When finalizing the choice of n for a pseudospectral method, an additional consideration is to choose a value that not only satisfies the tolerance conditions, but is also an efficient length for computing FFTs. In Mathematica, an efficient FFT does not require a power of two length since the Fourier command has a prime factor algorithm built in.

Typically, the difference order has a profound effect on the number of points required to satisfy the error estimate.

This makes a table of the number of points required to satisfy the a-priori error estimate as a function of the difference order.

A table of the number of points required as a function of difference order goes a long way towards explaining why the default setting for the method of lines is DifferenceOrder->4: the improvement from 2 to 4 is usually most dramatic and in the default tolerance range, fourth order differences do not tend to produce large roundoff errors, which can be the case with higher orders. Pseudospectral differences are often a good choice, particularly with periodic boundary conditions, but it is not a good choice for the default because they lead to full Jacobian matrices, which can be expensive to generate and solve if needed for stiff equations.

For non-periodic grids, the error estimate is done using only interior points. The reason is that the error coefficients for the derivatives near the boundary are different. This may miss features that are near the boundary, but the main idea is to keep the estimate simple and inexpensive since the evolution of the PDE may change everything anyway.

For multiple spatial dimensions, the determination is made one dimension at a time. Since better resolution in one dimension may change the requirements for another, the process is repeated in reverse order to improve the choice.

A-posteriori Error Estimates

When the solution of a PDE is computed with NDSolve, a final step is to do a spatial error estimate on the evolved solution and issue a warning message if this is excessively large.

These error estimates are done in a manner similar to the a-priori estimates described previously. The only real difference is that, instead of using grids with n and 2n points to estimate the error, grids with n/2 and n points are used. This is because, while there is no way to generate the values on a grid of 2n points without using interpolation, which would introduce its own errors, values are readily available on a grid of n/2 points simply by taking every other value. This is easily done in the Richardson extrapolation formula by using == 2 , which gives

This defines a function (based on functions defined in the previous section) that can compute an error estimate on the solution of the sine-Gordon equation from solutions for u and v expressed as vectors. The function has been defined to be a function of the grid since this is applied to a grid already constructed. (Note, as defined here, this only works for grids of even length. It is not difficult to handle odd length, but it makes the function somewhat more complicated.)

This solves the sine-Gordon equation with a Gaussian initial condition.

This is the grid used in the spatial direction which is the first set of coordinates used in the InterpolatingFunction. A grid with the last point dropped is used to obtain the values because of periodic continuation.

This makes a function that gives the a-posteriori error estimate at a particular numerical value of t.

This makes a plot of the a-posteriori error estimate as a function of t.

The large amount of local variation seen in this function is typical. For that reason, NDSolve does not warn about excessive error unless this estimate gets above 10 (rather than the value of 1, which is used to choose the grid based on initial conditions.) The extra factor of 10 is further justified by the fact that the a-posteriori error estimate is less accurate than the a-priori one. Thus, when NDSolve issues a warning message based on the a-posteriori error estimate, it is usually because new solution features have appeared or because there is instability in the solution process.

This is an example with the same initial condition used in the examples above, but where NDSolve gives a warning message based on the a-posteriori error estimate.

This shows a plot of the solution at t == 4. It is apparent that the warning message is appropriate because the oscillations near the peak are not physical.

When the NDSolve::eerr message does show up, it may be necessary for you to use options to control the grid selection process since it is likely that the default settings did not find an accurate solution.

Controlling the Spatial Grid Selection

The NDSolve implementation of the method of lines has several ways to control the selection of the spatial grid.

Tensor product grid options for the method of lines.

All of the options for tensor product grid discretization can be given as a list with length equal to the number of spatial dimensions, in which case the parameter for each spatial dimension is determined by the corresponding component of the list.

With the exception of pseudospectral methods on non-periodic problems, discretization is done with uniform grids, so when solving a problem on interval length , there is a direct correspondence between the Points and StepSize options:

The StepSize options are effectively converted to the equivalent Points values. They are simply provided for convenience since sometimes it is more natural to relate problem specification to step size rather then the number of discretization points. When values other then Automatic are specified for both the Points and corresponding StepSize option, generally, the more stringent restriction is used.

In the previous section an example was shown where the solution was not resolved sufficiently because the solution steepened as it evolved. The examples below will show some different ways of modifying the grid parameters so that the near shock is better resolved.

One way to avoid the oscillations that showed up in the solution as the profile steepened is to make sure that we use sufficient points to resolve the profile at its steepest. In the one-hump solution of Burgers` equation,

it can be shown [W76] that the width of the shock profile is proportional to ν as ν->0. More than 95% of the change is included in a layer of width 10 ν. Thus, if we pick a maximum step size of half the profile width, there will always be a point somewhere in the steep part of the profile, and there is a hope of resolving it without significant oscillation.

This computes the solution to Burgers' equation, such that there are sufficient points to resolve the shock profile.

Note that resolving the profile alone is by no means sufficient to meet the default tolerances of NDSolve, which requires an accuracy of . However, once we have sufficient point to resolve the basic profile, it is not unreasonable to project based on the a-posteriori error estimate shown in the NDSolve::eerr message (with an extra 10% since, after all, it is just a projection).

This computes the solution to Burgers' equation with the maximum step size chosen so that it should be small enough to meet the default error tolerances based on a projection from the error of the previous calculation.

To compare solutions like this, it is useful to look at a plot of the solution only at the spatial grid points. Because the grid points are stored as a part of the InterpolatingFunction, it is fairly simple to define a function that does this.

This defines a function that plots a solution only at the spatial grid points at a time t.

This makes a plot comparing the three solutions found at t = 4.

In this example, the left hand side of the domain really does not need so many points. The points need to be clustered where the steep profile evolves, so it might make sense to consider explicitly specifying a grid that has more points where the profile appears.

This solves Burgers' equation on a specified grid that has most of its points to the right of x = 1.

This makes a plot of the values of the solution at the assigned spatial grid points.

Many of the same principles apply to multiple spatial dimensions. Burgers' equation in 2 dimensions with anisotropy provides a good example.

This solves a variant of Burgers' equation in 2 dimensions with different velocities in the x and y directions.

This shows a surface plot of the leading edge of the solution a t = 2.

Similar to the 1 dimensional case, the leading edge steepens. Since the viscosity term (ν) is larger, the steepening is not quite so extreme, and this default solution actually resolves the front reasonably well. Therefore it should be possible to project from the error estimate to meet the default tolerances. A simple scaling argument indicates that the profile width in the x direction will be narrower than in the y direction by a factor of . Thus, it makes sense that the step sizes in the y direction can be larger than those in the x direction by this factor, or, correspondingly, that the minimum number of points can be a factor of 1/less.

This solves the 2 dimensional variant of Burgers' equation with appropriate step size restrictions in the x and y direction projected from the a-posteriori error estimate of the previous computation which was done with 69 points in the x direction.

This solution takes a substantial amount of time to compute, which is not surprising since the solution involves solving a system of more than 18000 ODEs. In many cases, particularly in more than 1 spatial dimension, the default tolerances may be unrealistic to achieve, so you may have to reduce them by using AccuracyGoal and/or PrecisionGoal appropriately. Sometimes, especially with the coarser grids that come with less stringent tolerances, the systems are not stiff and it is possible to use explicit methods, that avoid the numerical linear algebra which can be problematic, especially for higher dimensional problems. For this example, using Method->ExplicitRungeKutta gets the solution in about half the time.

Any of the other grid options can be specified as a list giving the values for each dimension. When only a single value is given, it is used for all the spatial dimensions. The two exceptions to this are MaxPoints, where a single value is taken to be the total number of grid points in the outer product, and Coordinates, where a grid must be specified explicitly for each dimension.

It is important to keep in mind that the a-posteriori spatial error estimates are simply estimates of the local error in computing spatial derivatives and may not reflect the actual accumulated spatial error for a given solution. One way to get an estimate on the actual spatial error is to compute the solution to very stringent tolerances in time for different spatial grids. To show how this works, consider again the simpler 1-dimensional Burgers` equation.

This computes a list of solutions using {33, 65, ..., 4097} spatial grid points to compute the solution to Burgers` equation for difference orders 2, 4, 6, and pseudospectral. The temporal accuracy and precision tolerances are set very high so that essentially all of the error comes from the spatial discretization. Note that by specifying {t, 4, 4} in NDSolve, only the solution at t = 4 is saved. Without this precaution, some of the solutions for the finer grids (which take many more time steps) could exhaust available memory. Even so, the list of solutions takes a substantial amount of time to compute.

Given two solutions, a comparison needs to be done between the two. To keep out any sources of error except for that in the solutions themselves, it is best to use the data that is interpolated to make the InterpolatingFunction. This can be done by using points common to the two solutions.

This defines a function to estimate error by comparing two different solutions at the points common to both. The arguments coarse and fine should be the solutions on the coarser and finer grids, respectively. This works for the solutions generated above with grid spacing varying by powers of 2.

To get an indication of the general trend in error (in cases of instability, solutions do not converge, so this does not assume that), you can compare the difference of successive pairs of solutions.

This defines a function that will plot a sequence of error estimates for the successive solutions found for a given difference order and uses it to make a logarithmic plot of the estimated error as a function of the number of grid points.

A logarithmic plot of the maximum spatial error in approximating the solution of Burgers` equation at t = 4 as a function of the number of grid points. Finite differences of order 2, 4, and 6 on a uniform grid are shown in red, green, blue, and magenta, respectively. Pseudospectral derivatives with uniform (periodic) spacing are shown in black.

The upper left part of the plot are the grids where the profile is not adequately resolved, so differences are simply of magnitude order 1 (it would be a lot worse if there was instability) However, once there are a sufficient number of points to resolve the profile without oscillation, convergence becomes quite rapid. Not surprisingly, the slope of the logarithmic line is -4, which corresponds to the difference order NDSolve uses by default. If your grid is fine enough to be in the asymptotically converging part, a simpler error estimate could be effected by using Richardson extrapolation as in the previous two sections, but on the overall solution rather than the local error. On the other hand, computing more values and viewing a plot gives a better indication of whether you are in the asymptotic regime or not.

It is fairly clear from the plot that the best solution computed is the pseudospectral one with 2049 points (the one with more points was not computed because its spatial accuracy far exceeds the temporal tolerances that were set). This solution can, in effect, be treated almost as an exact solution, at least up to error tolerances of or so.

To get a perspective of how best to solve the problem, it is useful to do the following: for each solution found that was at least a reasonable approximation, recompute it with the temporal accuracy tolerance set to be comparable to the possible spatial accuracy of the solution and plot the resulting accuracy as a function of solution time. The following (somewhat complicated) commands do this.

This identifies the "best" solution that will be used, in effect, as an exact solution in the computations that follow. It is dropped from the list of solutions to compare it to since the comparison would be meaningless.

This defines a function that, given a difference order, do, and a solution, sol, computed with that difference order, recomputes it with local temporal tolerance slightly more stringent than the actual spatial accuracy achieved if that accuracy is sufficient. The function output is a list of {number of grid points, difference order, time to compute in seconds, actual error of the recomputed solution}.

This applies the function to each of the previously computed solutions. (With the appropriate difference order!)

This removes the cases that were not recomputed and makes a logarithmic plot of accuracy as a function of computation time.

A logarithmic plot of the error in approximating the solution of Burgers` equation at t = 4 as a function of the computation time. Each point shown indicates the number of spatial grid points used to compute the solution. Finite differences of order 2, 4, and 6 on a uniform grid are shown in red, blue, and green, respectively. Pseudospectral derivatives with uniform (periodic) spacing are shown in black.

The resulting graph demonstrates quite forcefully that, when they work, as in this case, periodic pseudospectral approximations are incredibly efficient. Otherwise, up to a point, the higher the difference order, the better the approximation will generally be. These are all features of smooth problems, which, this particular instance of Burgers` equation is. However, the higher order solutions would generally be quite poor if we went towards the limit ν0.

One final point to note is that the above graph was computed using the Automatic method for the temporal direction. This uses LSODA which switches between a stiff and non-stiff method depending on how the solution evolves. For the coarser grids, strictly explicit methods are typically a bit faster, and, except for the pseudospectral case, the implicit BDF methods are faster for the finer grids. A variety of alternative ODE methods are available in NDSolve.

Error at the boundaries

The a-priori error estimates are computed in the interior of the computational region because the differences used there all have consistent error terms which can be used to effectively estimate the number of points to use. Including the boundaries into the estimates would complicate the process beyond what is justified for such and a-priori estimate. Typically, this approach is successful in keeping the error under reasonable control. However, there are a few cases which can lead to difficulties.

Occasionally it may occur that because the error terms are larger for the one-sided derivatives used at the boundary, NDSolve will detect an inconsistency between boundary and initial conditions which is an artifact of the discretization error.

This solves the one-dimensional heat equation with the left end held at constant temperature and the right end radiating into free space.

The NDSolve:ibcinc message is issued, in this case, completely to the larger discretization error at the right boundary. For this particular example, the extra error is not a problem because it gets damped out due to the nature of the equation. However, it is possible to eliminate the message by using just a few more spatial points.

This computes the solution to the same equation as above, but using a minimum of 50 points in the x-direction.

One other case where error problems at the boundary can affect the discretization unexpectedly is when periodic boundary conditions are given with a function which is not truly periodic, so that an unintended discontinuity is introduced into the computation.

This begins the computation of the solution to the sine-Gordon equation with a Gaussian initial condition and periodic boundary conditions. The NDSolve command is wrapped with TimeConstrained since solving the given problem can take a very long time and a large amount of system memory.

The problem here is that the initial condition is effectively discontinuous when the periodic continuation is taken into account.

This shows a plot of the initial condition over the extent of three full periods.

Since there is always a large derivative error at the cusps, NDSolve is forced to use the maximum number of points in an attempt to satisfy the a-priori error bound. To make matters worse, the extreme change makes solving the resulting ODEs more difficult, leading to a very long solution time which uses a lot of memory.

If the discontinuity is really intended, you will typically want to specify a number of points or spacing for the spatial grid which will be sufficiently to handle the aspects of the discontinuity you are interested in. To model discontinuities with high accuracy will typically take specialized methods which are beyond the scope of the general methods which NDSolve provides.

On the other hand, if the discontinuity was unintended, say in this example by simply choosing a computational domain which was too small, it can usually be fixed easily enough by extending the domain or by adding in terms to smooth things between periods.

This solves the sine-Gordon problem on a computational domain large enough so that the discontinuity in the initial condition is negligible compared to the error allowed by the default tolerances.