applet-magic.comThayer WatkinsSilicon Valley USA |
---|

for Finite Difference Schemes Which Use Noncentered Approximations for Derivatives in Differential Equations |

Forward and backward differences may be used to approximate the first derivative in a differential equation. Under the proper circumstances these finite difference approximations will give a reasonable approximation to the true solution to the differential equation but there are systematic deviations for both. These systematic errors have a simple explanation. The forward difference formula is best considered an approximation of the drivative at a half a step forward and the backwardward difference formula as an approximation of the derivative at a half step backward. Thus the forward and the backward approximations give approximation of the solution not to the original differential equation but to corresponding differential-difference equation. For example, consider the differential equation

dy/dt = cy

which has the true solution of

y(t) = y(0)exp[ct]

The forward finite difference approximation is

(y

where h is the time step Δt and y_{i}=y(ih).

But

(y

so the finite difference scheme is an approximation to the solution of the differential-difference equation

dy/dt|

which is equivalent to

dy/dt = y(t-h/2)

Differential-difference equations (DDE's) are more complex than differential equations but
for simple, linear equations the standard technique for solving differential equations
also works. Consider a solution to the differential-difference equation of the
form

y(t) = y_{0}exp[kt]. This form substituted into the
differential-difference
equation yields:

ky

= cy

which reduces to

k = c

This condition for k is a transcendental equation. It cannot be solved analytically,
but its solution can be found to any specified degree of accuracy easily by
numerical methods. For sufficiently small values of h the
exponential function exp[-kh/2] is closely approximated by

(1-kh/2). Thus the solution for k is the solution to
the equation

k = c(1-kh/2)

which is

k = c/(1+ch/2)

which is approximately

k = c(1-ch/2)

Thus the growth rate k of the solution of the corresponding differential-difference equation is less than that of the differential equation c. In the case c is positive k is c multiplied by a factor which is less than unity. If c is negative then k is c multiplied by a factor which is greater than unity so k is more negative than c. Since the solution to the finite difference approximation is a closer approximation to the differential-difference equation the finite difference approximation for c>0 is less than the true solution. For c<0 the finite difference approximation will also be below the true solution.

The backward difference approximation of the first derivative

(y

is an approximation of the differential-difference equation

dy/dt|

which is equivalent to

dy/dt = y(t+h/2)

A solution of the form y(t) = y_{0}exp[kt] leads to the condition which k
must satisfy of

k = c

which for small values of h reduces to

k = c(1+kh/2)

which has the solution

k = c(1+ch/2)

Thus for positive values of c the backward difference formula leads to growth rates greater than that of the true solution.

Below is shown the true solutions of the differential equation and the two differential-difference equations for c=1 and h=0.1.

When the numerical solutions using the forward and backward are displayed in the diagram it is obvious that those values are tracking the corresponding DDE's rather than the differential equation. The fact that the forward and and backward difference schemes are reasonable approximations for the solution to the differential equation is based solely on the closeness of the solutions to the DDE's to the solution to the differential equation.

The situation for the case in which the coefficient in the differential equation is negative is shown below. The relationships of the forward and backward difference schemes to the solutions to the corresponding DDE's and the differential equation are the same as in the case of a positive coefficient in the differential equation.