Question

Based on the Laplacian 9-point stencil, define a symmetric 9-point stencil numerical method for Helmholtz equation (1) and do the following: (a) Determine the truncation error in terms of f and u only, in a similar way as done in question (1). However, for this 9-point stencil, notice that is possible to eliminate all the derivatives of u from the leading term of the truncation error. (b) Use part (3a) to construct a 9-point fourth order numerical method for the BVP defined in question 2a. Show (analytically) that the new finite difference scheme is actually fourth order.

Answer 1

A symmetric 9-point stencil method for the Helmholtz equation is constructed by determining a finite difference approximation that eliminates lower order terms in the truncation error. Demonstrating that the scheme is fourth order involves showing that the leading terms of the error are canceled, with only higher-order terms remaining.

Step-by-step explanation:

The question relates to the development of a symmetric 9-point stencil numerical method for solving the Helmholtz equation and evaluating its truncation error. To construct such a method, a finite difference approximation involving points in a grid surrounding a central point is used. By expanding the finite difference equations and matching terms with the Taylor series expansion of the differential equation, the truncation error can be determined. To achieve a fourth-order method, the coefficients of the stencil can be adjusted such that the leading terms of the error cancel out, leaving only higher-order terms.

To analytically demonstrate that the scheme is fourth order, one would show that fourth-order derivatives remain while the second and third-order derivatives, which contribute to the leading term of the truncation error, are eliminated through the particular coefficients used in the 9-point stencil. Additional verification might involve calculating the local truncation error and ensuring that it is on the order of &O(h^4), where h is the grid spacing.