Truncation error, finite difference approximation

Comparing this equation to a finite difference approximation (Eqn. 20.29), we see that in the numerical solution we carry only the first term in the series, the d Q /3x term, omitting the higher order entries. The Taylor series is truncated, then, and the resulting error called truncation error. 

By adding the forward (explicit) finite-difference approximation to each side of this equation, we can identify both the explicit Euler algorithm and an expression for the local truncation error ... 

The consistency of a finite difference approximation is the behavior of this representation when the mesh is refined. In a one dimensional case, for example, the mesh will indicate the value for Ax, which, as we discussed above, dictates the value of the truncation error. Thus, a finite difference representation of a PDE is said to be consistent if the truncation error goes to zero as the grid size (or Ax) goes to zero. 

In solving the finite-difference approximation, we let FDA = 0 and, in fact, do not solve the differential equation, but rather the difference between the ODE and TE. For example, in Table 4.3, the deviation of the numerical results from those of the exact solution is caused by the truncation error, since Ax is not small enough to eliminate the effect of the truncated terms. 

Equation (4.70) indicates that the governing partial differential equation (PDE) equals the finite-difference approximation (FDA) to within a truncation error (TE) which is of 0[Af, (Ax)2]. Clearly, the first-order forward difference has only first-order accuracy [recall Eq. (4.43)]. However, differentiating BT/Bt = ud2T/dx2 twice with respect to x and once with respect to f, respectively, shows that the first two terms of TE can be combined into one,... 

The existence of truncation errors in finite difference approximations to differential equations is discussed in numerical analysis texts with respect to round-off error and computational instabilities (Roache, 1972 Richtmyer and Morton, 1957), but Lantz (1971) was among the first to address the form of the truncation error as it related to diffusion. Lantz considered a linear, convective, parabolic equation similar to 9u/9t + U 9u/9x = e S u/Sx and differenced it in several ways. He showed that the effective diffusion coefficient was not 8, as one might have suggested analytically, but 8 + 0(Ax, At) (so that the actual diffusion term appearing in computed solutions is the modified coefficient times c2u/9x2) where the 0(Ax,At) truncation errors, being functions of u(x,t), are comparable in magnitude to 8. Because this artificial diffusion necessarily differs from the actual physical model, one would expect that the entropy conditions characteristic of the computed results could likely be fictitious. 

The order of a numerical scheme is determined by the order of the truncation error, which is defined as the difference between the exact solution and the finite-difference approximation. Higher-order approximations may also be derived. For example, the fourth-order approximation in space combined with the second-order approximation in time has become popular for simulating ground motion (e.g., Olsen 1994 Graves 1996). 

As we can imagine, most of these issues are directly related to the order of the approximation used in the finite difference representation. In fact, the truncation error (as shown in Chapter 7) is the difference between the PDE and the FD representation, which is represented by the terms collapsed in 0(Axn). For problems represented by PDEs with more than one independent variable the truncation error will be the sum of the truncation error for each FD representation. For example, for a transient one dimensional PDE, where we use a first order approximation for the time derivative and a second order for the spatial derivative, we will have a truncation error that is O(At) + 0 Ax2), which can also be written as 0(At, Ax2). 

Up to this point, we have discussed how to develop finite difference expressions for PDE s with any desired level of accuracy. We expect that the truncation error, the consistency and stability will be improved as the order of the approximation is increased. However, we have not mentioned the connection between the physics of the problem under consideration, and represented by the PDE, and the way we select the points for the finite difference expression, i.e., i — 2, i — 1, i + 1, or i + 2, as well as the order of the approximation, i.e., first order, second order, etc. In other words, the points, order and grid size are not arbitrarily chosen to approximate the derivatives at a point i, but should be chosen according to the physical principles of the problem. This concept is referred to as info-travel and will be discussed here. 

A consistent numerical scheme produces a system of algebraic equations which can be shown to be equivalent to the original model equations as the grid spacing tends to zero. The truncation error represents the difference between the discretized equation and the exact one. For low order finite difference methods the error is usually estimated by replacing all the nodal values in the discrete approximation by a Taylor series expansion about a single point. As a result one recovers the original differential equation plus a remainder, which represents the truncation error. 

The derivative (D) being approximated by the finite-difference operator (FD) to within a truncation error (TE) (or, discretization error). The foregoing mathematical consideration provides an estimate of the accuracy of the discretization of the difference operators. It shows that TE is of the order of (Ax)2 for the central difference, but only O(Ax) for the forward and backward difference operators of first order. Equations (4.41) and (4.42) involve 2 or 3 nodes around node i at x , leading to 2- and 3-point difference operators. Considering additional Taylor series expansions extending to nodes i + 2 and i - 2 etc., located at x + 2Ax and x. — 2Ax, etc., respectively, one may derive 4- and 5-point difference formulas with associated truncation errors. Results summarized in Table 4,8 show that a TE of O(Ax)4 can be achieved in this manner. The penalty for this increased accuracy is the increased complexity of the coefficient matrix of the resulting system of equations. 

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