On Multiscale Variational and Streamline Diffusion Schemes for a Coupled Nonlinear Telegraph System

ABSTRACT

We propose a hybrid of streamline diffusion (SD) method and variational multiscale scheme (VMS) for approximate solution of a coupled nonlinear system of telegraph equations. The reason for using multiscale scheme is due to the fact that, compared to the evolved system, in the primary time iteration steps higher degree schemes in spatial variable are not necessary. On the other hand, the multiscale strategy may be a viewed as a particular type of adaptivity where different scales play the role of coarse or fine refinement procedures. In this setting, certain data and geometric singularities that are better studied through an adaptive approach, which here is replaced by a multiscale scheme. We prove stability estimates and derive optimal convergence rates due to the maximal available regularity of the exact solution. This study concerns both theoretical as well as some numerical aspects. The theoretical part, mainly, concerns the stability and convergence issues whereas in the numerical part, we deal with the construction and of the discretization multiscale schemes. The results are justified through some numerical implementations where, in particular by the constructed multiscale scheme, one may circumvent the above mentioned trouble with the primary time steps.

KEYWORDS:

• Telegraph equation
• variational multiscale scheme
• discontinuous Galerkin
• streamline diffusion
• stability
• convergence