A positivity preserving convergent event based asynchronous PDE solver
This work offers a new adaptive numerical method for conservation laws, potentially improving efficiency for problems with localized features, though it is an incremental contribution to the field of numerical PDE solvers.
The paper introduces an asynchronous event-based numerical scheme for conservation equations that updates only two cells per event, leading to adaptive space-time resolution. Numerical results demonstrate that the error decreases to zero as a control parameter is reduced, and a convergence proof framework is provided using matrix exponentials and the BCH formula.
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.