A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation
This work provides a numerical scheme preserving asymptotic behavior for a specific PDE, which is incremental for the field of numerical analysis.
The authors analyze the large-time behavior of the augmented Burgers equation, proving well-posedness and decay rates, and propose a semi-discrete scheme that preserves the asymptotic behavior. Numerical experiments confirm accuracy.
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 $L^1$-$L^p$ decay rates. The asymptotic behavior of the solution is obtained by showing that the influence of the convolution term $K*u_{xx}$ is the same as $u_{xx}$ 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.