Application of a perfectly matched layer to the nonlinear wave equation |
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| Affiliation: | 1. Center for Applied and Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA;2. Department of Information Technology, Division of Scientific Computing, Uppsala University, Uppsala S-75105, Sweden;1. School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom;2. Department of Materials Science and Technology, University of Crete, 71003, Heraklion, Crete, Greece;3. FORTH-IESL, PO Box 1385, 71110, Heraklion, Crete, Greece;4. Spin Optics Laboratory, St-Petersburg State University, 1, Ulianovskaya, 198504, St-Petersburg, Russia;5. Quantum Center, 143025, Skolkovo, Moscow Region, Russia;1. Central Aerohydrodynamic Institute, Zhukovskii, Russia;2. V. A. Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia;3. Moscow Institute of Physics and Technology, Dolgoprudnyi, Russia;1. Lab of Digital Image and Intelligent Computation, Shanghai Maritime University, Shanghai 201306, China;2. Department of Neurology, Shanghai Jiao Tong University Affiliated Sixth People''s Hospital, Shanghai 200233, China;1. Department of Mechanical and Manufacturing Engineering, Aalborg University, Fibigerstraede 16, 9220 Aalborg, Denmark;2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, United Kingdom |
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| Abstract: | We derive a perfectly matched layer-like damping layer for the nonlinear wave equation. In the layer, only two auxiliary variables are needed. In the linear case the layer is perfectly matched, but in the nonlinear case it is not. Well posedness is established for the linear case. We also prove various energy estimates which can be used as a starting point for establishing stability of more general cases. In particular, we are able to show estimates for a special type of nonlinearity.Numerical experiments that show the effectiveness of the layer are presented both for nonlinear and linear problems. In the computations, we use an eighth order summation-by-parts discretization in space and implement the boundary conditions using a penalty procedure. We present new stability results for this discretization applied to the second order wave equation in the case with Dirichlet boundary conditions. |
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