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Scheifinger, Malik [Institut für Angewandte und Numerische Mathematik]
Code to "Strong norm error bounds for quasilinear wave equations under weak CFL-type conditions" (v2)
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(Abstract) In the present paper we consider a class of quasilinear wave equations on a smooth, bounded domain. We discretize it in space with isoparametric finite elements, and apply a semi-implicit Euler and midpoint rule as well as the exponential Euler and midpoint rule to obtain four fully discrete schemes. We derive rigorous error bounds of optimal order for the semi-discretization in space and the fully discrete methods in norms which are stronger than the classical $H^1\times L^2$ energy norm under weak CFL-type conditions. To confirm our theoretical findings, we also present numerical experiments.
(Technical Remarks) This code is used for the numerical experiment in Section 2 of the paper "Strong norm error bounds for quasilinear wave equations under weak CFL-type conditions" by Benjamin Dörich. Part of the code relies on code written by J. Leibold in . The computations are done in C++ using the Finite Element library deal.II; the plots then are generated with Python3. To use this code, deal.II (release 9.5.1) has to be installed, cf. In order to compile the program, open a terminal session in this folderand call "cmake -DDEAL_II_DIR=/path/to/deal.II ." Next, call "make release" and "make". Then, one can run the commands (can be done in paralell) ./main P1 ./main P2 ./main P3 ./main P3_half ./main euler ./main midpoint ./main alpha60P1 ./main alpha60P2 ./main alpha80P1 ./main alpha80P1 ./main alpha90P1 ./main alpha90P2 ./main alpha95P1 ./main alpha95P2 to execute the code. This performs the computations and generates the files parameters_P1.ini_space_Q1.txt parameters_P2.ini_space_Q2.txt parameters_P3.ini_space_Q3.txt parameters_P3_half.ini_space_Q3.txt parameters_euler.ini_time_ImplEuler parameters_midpoint.ini_time_MidpointRule parameters_alpha_60_P1.ini_space_Q1.txt parameters_alpha_60_P1.ini_space_Q2.txt parameters_alpha_80_P1.ini_space_Q1.txt parameters_alpha_80_P2.ini_space_Q2.txt parameters_alpha_90_P1.ini_space_Q1.txt parameters_alpha_90_P2.ini_space_Q2.txt parameters_alpha_95_P1.ini_space_Q1.txt parameters_alpha_95_P2.ini_space_Q2.txt in the folder "error" containing the results of the numerical experiments. After that, the plots can be generated with the Python3 Script using in the terminal python3 in the folder "tikz".
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Scheifinger, Malik
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