Comparison of some fourth order difference schemes for hyperbolic problems

Abstract

Our purpose is the comparison of a few difference schemes for application to hyperbolic equations such as those found in computational fluid dynamics. In order to provide a valid comparison we will use the optimal time-step @Dt and mesh spacing @Dx to achieve a preassigned accuracy. This means we assign an error in the solution to the equation, usually 5% or 0.5% relative error, and then determine the values of @Dt and @Dx to yield this error with minimal computational effort. In practice one cannot be this precise; however, this is the only reasonable way to compare the schemes. We will describe a graphical means to determine these optimal values for test problems for which the exact solution is known. We will also introduce two predictor-corrector schemes which, to our knowledge, have not previously been applied to hyperbolic equations. In addition to cost effectiveness a scheme should be evaluated on ''robustness'', that is, the ability to handle shocks, stiff equations, discontinuous coefficients, and a variety of boundary conditions. Perhaps problems involving the first three features will require special schemes tailored to that type of difficulty; however, a scheme should be able to handle a variety of boundary conditions subject only to the condition that these be properly posed. The leapfrog schemes are especially prone to boundary instabilities as the work of Oliger, Gustafsson, Elvius, and Sundstrom has shown.^7^,^8^,^9 Therefore, we also test some of these schemes for boundary instability. We test the leapfrog scheme using fourth order spatial differences, a predictor-corrector scheme based on the Milne corrector, another based on the implicit Runge-Kutta corrector, a standard Runge-Kutta scheme, and a variable time-step Runge-Kutta-Fehlberg scheme.^2^,^5