Spatially partitioned embedded Runge-Kutta methods (Q2870632)

The paper deals with spatially partitioned embedded Runge-Kutta (SPERK) methods as an appropriate alternative to more traditional schemes in the time integration of the ordinary differential equation (ODE) resulting from a previous space discretization of partial differential equations (PDEs) when the smoothness of their solution or the magnitude of their coefficients vary strongly in space. The attention is focused on embedded pairs, since they offer a better efficiency and avoid the order reduction which is present in other non-embedded schemes.NEWLINENEWLINEWith SPERK integrators, each component method of the overall scheme is applied over different parts of the spatial domain, and this allows, in particular, to use larger time steps. SPERK methods are applied in two different ways. In the first (the equation-based partitioning), the ODE is partitioned, and at each point in space a particular time-stepping scheme is selected. In this case, the order of accuracy is shown to be equal to the minimum of the order of accuracy of the schemes composing the embedded pair, but on the other hand it is not conservative when applied to a scalar conservation law.NEWLINENEWLINEThe second partitioning method operates on fluxes, and is called flux-based partitioning. This has the advantage of being conservative when applied to a conservation law. It is shown that it may suffer from order reduction, although this loss of accuracy is not observed in practice.NEWLINENEWLINEBoth procedures are positivity preserving under a suitable time step restriction when the component schemes are strong stability preserving methods.NEWLINENEWLINESeveral explicit SPERK schemes are designed and numerically illustrated on numerical examples. One of them, in particular, is suitable to be used in the time integration of PDEs, previously discretized in space with fifth-order weighted non-oscillatory techniques.