Component splitting for semi-discrete Maxwell equations

Abstract

A time-integration scheme for semi-discrete linear Maxwell equations is proposed. Special for this scheme is that it employs component splitting. The idea of component splitting is to advance the greater part of the components of the semidiscrete system explicitly in time and the remaining part implicitly. The aim is to avoid severe step size restrictions caused by grid-induced stiffness emanating from locally refined space grids. The proposed scheme is a blend of an existing secondorder composition scheme which treats wave terms explicitly and the second-order implicit trapezoidal rule. The new blended scheme retains the composition property enabling higher-order composition.

Composition Methods, Maxwell's Equations, and Source Terms

“…4 ) As proved in [1,13], this local order reduction does not affect the 2nd-order convergence of method (7) for τ ∼ h, h → 0.…”

“…For the Maxwell equations it has for example been studied in [9] and [1,13]. Our error analysis concerns temporal convergence towards the true solutions of the underlying PDE problem restricted to the space grid.…”

“…In [1,13] the following theorem has been proved: This theorem thus says that the second-order method does not suffer from order reduction. This second-order result is special in that the local error may suffer from reduction, cf.…”

“…Below we will review the local errors for τ ∼ h, h → 0 since we will need these when the method is used as building block for the higher-order composition methods. 2) For the full proof of the theorem explaining the fortunate cancelation we refer to [1,13]. Details on stability properties and energy conservation can also be found in [1].…”

Multi-level explicit local time-stepping methods for second-order wave equations

“…Moreover, even when each individual method has order two, the implicit-explicit component splitting can reduce by one the overall space-time convergence rate of the resulting scheme [6,7]. Recently, Descombes, Lanteri and Moya [7] remedied that unexpected loss in accuracy and hence recovered second-order convergence, by using the LF/CN-IMEX approach of Verwer [8] instead, yet at the price of a significantly larger albeit sparse linear system. In contrast, locally explicit time-stepping methods remain fully explicit by taking smaller time-steps in the "fine" region, that is precisely where the smaller elements are located.…”