Accuracy and Resolution in the Computation of Solutions of Linear and Nonlinear Equations

Abstract

This chapter describes the accuracy and resolution in the computation of solutions of linear and nonlinear equations. In many problems, one is presented with piecewise initial data whose discontinuities occur along surfaces. According to the theory of hyperbolic equations, solutions with such initial data are themselves piecewise with their discontinuities occurring across characteristic surface. At points away from the discontinuities, the truncation error is small. In these regions, it is reasonable to use difference approximations of high order accuracy, except for the danger that the large truncation error at the discontinuities propagates into the smooth region. According to the theory of hyperbolic conservation laws, solutions of systems of the form are in general discontinuous. The discontinuities, called shocks, need not be present in the initial values but arise spontaneously and their speed of propagations is governed by the Rankine–Hugoniot jump relation. It is more difficult to construct accurate approximations of discontinuous solutions of nonlinear equations than of linear equations.