Abstract
We introduce a hybrid method to couple continuous Galerkin finite element methods and high-order finite difference methods in a nonconforming multiblock fashion. The aim is to optimize computational efficiency when complex geometries are present. The proposed coupling technique requires minimal changes in the existing schemes while maintaining strict stability, accuracy, and energy conservation. Results are demonstrated on linear and nonlinear scalar conservation laws in two spatial dimensions.
Highlights
Finite element (FE) methods and finite difference (FD) methods are among the most studied and employed numerical methods for partial differential equations (PDEs), which serve numerous industrial applications and other research areas, e.g., economics, engineering, chemistry, physics
We show that even when considering nonlinear conservation laws, nondiagonal norm matrices, namely the continuous FE mass matrices, could be applicable without further techniques involved, e.g., mass lumping [31], to the classical Galerkin formulation
We prove that the following FD–FE coupling SATs impose stability to (26)
Summary
Finite element (FE) methods and finite difference (FD) methods are among the most studied and employed numerical methods for partial differential equations (PDEs), which serve numerous industrial applications and other research areas, e.g., economics, engineering, chemistry, physics. A well-designed numerical framework for energy stable approximations of time-dependent problems is the combination of the summation-by-parts (SBP) operators, see Kreiss and Scherer [15], and the simultaneous-approximationterm (SAT) technique, see Carpenter et al [4]. The coupling of mixed order schemes on nonconforming grids was, for the first time, proposed in an energy stable manner by Mattsson and Carpenter [22]. Fundamental properties of SBP schemes are extended to blockto-block coupling: strict stability, accuracy, and conservation This technique paves the way for many possibilities to couple different methods which can be written in SBP form. The proposed technique results in a provably stable, accurate, and energy-conserved unified SBP method. Interface continuity is weakly imposed by the SAT technique using nondiagonal norm SBP-preserving interpolation operators. Two approaches to constructing the required interpolation operators are shown in Appendix A
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