Higher order monotonic convective difference schemes

Abstract

Eulerian mesh computations of simple linear and nonlinear one-dimensional wave propagations with second and higher order convective difference schemes exhibit nonphysical oscillations near steep gradient regions of the solution. This computational noise may propagate a significant number of mesh intervals away from the regions in which it is generated and severely compromise the solution accuracy. Boris, Book, and van Leer have developed nonlinear filtering techniques for second order convection algorithms which effectively remove computational noise from steep gradient solutions. The principal disadvantage of these filtering techniques is that solutions of sharply peaked waves are penalized in amplitude accuracy of the extrema. A nonlinear method of filtering computational noise from fourth and higher order accurate convective difference schemes is introduced which removes the computational noise without inflicting significant amplitude losses in sharply peaked waves. One-dimensional simple linear and nonlinear test problems are used to illustrate the performance of various unfiltered and filtered convective difference schemes. It is noted that the filtered higher order convective difference schemes require less than one-third of the mesh points of the filtered second order convective difference schemes to model the extrema of sharply peaked waves to the same accuracy. Finally, the Accurate Space Derivative method of Gazdag is shown to function with the global numerical differentiation performed with compact polynomial splines. This method is at least sixth and tenth order accurate, respectively, for modeling linear waves with cubic and quintic spline differentation for Courant numbers less than about 1 4 .