Preprint A351/2004
Maximum Norm Stability of Difference Schemes for Parabolic Equations on Overset Non-Matching Space-Time Grids

Giovanni Russo | Mathew, Tarek

**Keywords: **
Non-matching overset space-time grids | maximum norm stability | composite grids | parallel Schwarz alternating method | parabolic equations | discrete maximum principle | discrete barrier functions

In this paper, theoretical results are described on the maximum norm stability and
accuracy of finite difference discretizations of parabolic equations on overset nonmatching space time grids. We consider parabolic equations containing a linear reaction term, on a space-time domain decomposed into an overlapping collection of cylindrical space-time subregions. Each of the space-time subregions is assumed to be independently grided with local mesh and time step sizes. The different sapce-time grids need not match on the regions of overlap,
and the time steps can vary across subregions. On each space-time region, the
parabolic equation may be discretized by an implicit or explicit $\theta$-scheme,
satisfying a discrete maximum principle. The local discretizations are coupled globally, without the use of Lagrange multipliers. Maximum norm stability and
optimal order convergence bounds are proved. Schwarz type iterative algorithms
are described for solving the large system of algebraic equations resulting from such discretizations. Convergence bounds are derived using a Picard contraction mapping principle.
Applications are indicated to reaction diffusion equations and to parabolic-hyperbolic approximations of parabolic equations.