A positivity preserving convergent event based asynchronous PDE solver

TL;DR

This paper introduces a novel asynchronous PDE solver that preserves positivity, adapts in space and time, and converges as the control parameter decreases, with potential for theoretical analysis via matrix exponential techniques.

Contribution

It presents a new asynchronous, positivity-preserving PDE solver with convergence proof framework based on matrix exponential methods.

Findings

Error decreases to zero as control parameter is reduced

Scheme is positivity preserving and adaptive in space and time

Convergence can be analyzed using matrix exponential techniques

Abstract

A new numerical scheme for conservation equations based on evolution by asynchronous discrete events is presented. During each event of the scheme only two cells of the underlying Cartesian grid are active, and an event is processed as the exact evolution of this subsystem. This naturally leads to and adaptive scheme in space and time. Numerical results are presented which show that the error of the asynchronous scheme decreases to zero as a control parameter is reduced. The construction of the scheme allows it to be expressed as repeated multiplications of matrix exponentials on an initial state vector; thus techniques such as the Goldberg series and the Baker Campbell Hausdorff (BCH) formula can be used to explore the theoretical properties of the scheme. We present the framework of a convergence proof in this manner.