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A Non-stiff Summation-By-Parts Finite Difference Method for the Scalar Wave Equation in Second Order Form: Characteristic Boundary Conditions and Nonlinear Interfaces
Brittany A Erickson, Jeremy E Kozdon, Tobias W Harvey
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Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard method can become arbitrarily stiff when characteristic boundary conditions and nonlinear interface conditions are used. Here we propose a new technique that avoids this stiffness by using characteristic variables to "upwind" the boundary and interface treatment. This is done through the introduction of an additional block boundary displacement variable. Using a unified energy, which expresses both the standard as well as characteristic boundary and interface treatment, we show that the resulting scheme has semidiscrete energy stability for the scalar anisotropic wave equation. The theoretical stability results are confirmed with numerical experiments that also demonstrate the accuracy and robustness of the proposed scheme. The numerical results also show that the characteristic scheme has a time step restriction based on standard wave propagation considerations and not the boundary closure.
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