Error Analysis of Explicit Partitioned Runge–Kutta Schemes for Conservation Laws

Abstract

An error analysis is presented for explicit partitioned Runge-Kutta methods and multirate methods applied to conservation laws. The interfaces, across which different methods or time steps are used, lead to order reduction of the schemes. Along with cell-based decompositions, also flux-based decompositions are studied. In the latter case mass conservation is guaranteed, but it will be seen that the accuracy may deteriorate.

Multirate time stepping for accelerating explicit discontinuous Galerkin computations with application to geophysical flows

“…terms of the TV norm [15], the RK2aC scheme should theoretically behave better for problems that develop shocks because it inherits the monotonicity properties [21,22] of the base method [14,27]. Note that first-order time-stepping schemes are too dissipative and therefore inappropriate.…”

“…If 1 2 t is stable for 0 , then t may be assumed stable for i>0 . Hundsdorfer et al [27] discuss, within the framework of partitioned RK methods, the defects of multirate methods of first and second orders because of either the local inconsistency or the lack of mass conservation. For a large value of N , the number of elements in the mesh, one can show that a speedup of 2 is obtained compared with the same s-stage ERK method applied with the same time step t 2 everywhere.…”

“…However, if we consider a partitioning based on fluxes rather than in terms of the components, mass preservation is guaranteed at any stage of the explicit PRK method [15,27]. Because c G D c F , the solutions at each inner stage of the method are all computed at the same intermediate time steps Q t k i .…”

“…The drawback, however, is that the resulting schemes are not mass preserving. Hundsdorfer et al [27] discuss, within the framework of partitioned RK methods, the defects of multirate methods of first and second orders because of either the local inconsistency or the lack of mass conservation. They give a particular attention to monotonicity properties of the considered multirate schemes.…”

“…A drawback of the method is that there is no conservation of the fluxes at the critical interface because b G ¤ b F . However, if we consider a partitioning based on fluxes rather than in terms of the components, mass preservation is guaranteed at any stage of the explicit PRK method [15,27]. This approach has not been investigated in this paper.…”