The numerical behavior of high-order finite difference methods

We investigate the numerical behavior of a fourth-order accurate method. The results are compared with those of a second-order method. We have verified the rate of convergence for the numerical solution. As test cases the simple hyperbolic model equationu₁+u_x=0 and the two-dimensional Euler equations over backward-facing step have been used. The fourth-order method has been implemented on a dataparallel computer, and the difference operators have been designed to minimize the bandwidth. We also derive boundary modified, semidefinite artificial viscosity operators of arbitrary order of accuracy. The viscosity operators are presented in a form that is particularly well-suited for the implementation on dataparallel computers.