A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation

In this paper we analyze the large-time behavior of the augmented Burgers equation. We first study the well-posedness of the Cauchy problem and obtain L1-Lp decay rates. The asymptotic behavior of the solution is obtained by showing that the influence of the convolution term K ∗ uxx is the same as uxx for large times. Then, we propose a semi-discrete numerical scheme that preserves this asymptotic behavior, by introducing two correcting factors in the discretization of the non-local term. Numerical experiments illustrating the accuracy of the results of the paper are also presented.