Theoretical stability and accuracy analysis of ADER-ZIST

Establish a theoretical analysis of the stability and accuracy of the ADER-ZIST mixed explicit ADER-DG and zonally implicit space-time DG scheme, at least for linear problems in one space dimension.

Background

ADER-ZIST is presented as a mixed explicit–implicit discontinuous Galerkin method for hyperbolic conservation laws on Cartesian cut-cell meshes. The numerical experiments indicate that the method can retain its nominal order of accuracy and remain stable with time steps determined by the Cartesian background mesh rather than by arbitrarily small cut cells.

The paper does not provide a theoretical stability or convergence analysis. The authors explicitly identify such an analysis—initially for linear one-dimensional problems—as a future unresolved task, with broader extensions to three-dimensional problems, moving embedded boundaries, more general hyperbolic systems, and space-time adaptive mesh refinement also listed as directions for future work.