Optimal Prediction of Burgers’s Equation

Abstract

We examine an application of the optimal prediction framework to the truncated Fourier–Galerkin approximation of Burgers’s equation. Under particular conditions on the density of the modes and the length of the memory kernel, optimal prediction introduces an additional term to the Fourier–Galerkin approximation which represents the influence of an arbitrary number of small wavelength unresolved modes on the long wavelength resolved modes. The modified system, called the t‐model by previous authors, takes the form of a time‐dependent cubic term added to the original quadratic system. Numerical experiments show that this additional term restores qualitative features of the solution in the case where the number of modes is insufficient to resolve the resulting shocks (i.e., zero or very small viscosity) and for which the original Fourier–Galerkin approximation is very poor. In particular, numerical examples are shown in which the kinetic energy decays in the same manner as in the exact solution, i.e., as $t^...