Transformed implicit-explicit DIMSIMs with strong stability preserving explicit part
For scientists and engineers solving differential equations with stiff and non-stiff parts, this work offers new numerical methods that combine SSP and stability, but it is incremental as it extends existing IMEX DIMSIM frameworks.
The authors construct implicit-explicit DIMSIMs where the explicit part has strong stability preserving (SSP) property and the implicit part is A- or L-stable, providing examples up to order four with good stability properties.
For many systems of differential equations modeling problems in science and engineering, there are often natural splittings of the right hand side into two parts, one of which is non-stiff or mildly stiff, and the other part is stiff. Such systems can be efficiently treated by a class of implicit-explicit (IMEX) diagonally implicit multistage integration methods (DIMSIMs), where the stiff part is integrated by implicit formula, and the non-stiff part is integrated by an explicit formula. We will construct methods where the explicit part has strong stability preserving (SSP) property, and the implicit part of the method is $A$-, or $L$-stable. We will also investigate stability of these methods when the implicit and explicit parts interact with each other. To be more precise, we will monitor the size of the region of absolute stability of the IMEX scheme, assuming that the implicit part of the method is $A$-, or $L$-stable. Finally we furnish examples of SSP IMEX DIMSIMs up to the order four with good stability properties.