Arbitrary Dimension Convection-Diffusion Schemes for Space-Time Discretizations

TL;DR

This paper introduces a novel approach by embedding time-dependent PDEs into convection-diffusion equations in higher dimensions, analyzing two discretization schemes, and extending the EAFE scheme to arbitrary order with numerical validation.

Contribution

It presents a new method for space-time discretization of PDEs via embedding into convection-diffusion equations and extends the EAFE scheme to arbitrary order.

Findings

Both schemes are stable and accurate for the proposed embedding.

The extended EAFE scheme achieves arbitrary order accuracy.

Numerical results confirm the feasibility of the approach.

Abstract

This note proposes embedding a time dependent PDE into a convection-diffusion type PDE (in one space dimension higher) with singularity, for which two discretization schemes, the classical streamline-diffusion and the EAFE (edge average finite element) one, are investigated in terms of stability and error analysis. The EAFE scheme, in particular, is extended to be arbitrary order which is of interest on its own. Numerical results, in combined space-time domain demonstrate the feasibility of the proposed approach.