Comparison of some fourth order difference schemes for hyperbolic problems

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 Δt and mesh spacing Δx 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 Δt and Δx 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.