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High-order discretizations for the wave equation based on the modified equation technique

Cyril Agut 1, 2, * Julien Diaz 1, 2 Abdelaâziz Ezziani 1, 2
* Corresponding author
1 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau], Inria Bordeaux - Sud-Ouest
Abstract : The accurate solution to the wave equation implies very high computational burdens, even when using high-order space discretization methods. Besides, if we use explicit time discretization methods (such as the classical Leap-Frog scheme), the time step has to satisfy a CFL (Courant-Friedrichs-Lewy) condition to ensure the stability of the scheme. The smaller the space step is, the smaller the CFL condition and the higher the number of iterations will be. To improve the accuracy of the Leap-Frog scheme, we may consider the modified equation technique, which allows to obtain explicit arbitrary 2p-th order scheme in time. The price to pay is p matricial multiplications at each time step when the Leap-Frog scheme only requires one, whereas the CFL condition is multiplied by $\alpha_p>1 $. For p=2 (fourth-order scheme), $\alpha_2=1.7$, so that the additional computational cost is small, but for higher-order scheme the increase of the CFL condition is generally not sufficient to counterbalance the number of additional multiplications. Recently, a technique has been proposed to optimize the coefficients $\alpha_p$, but it requires more matricial multiplications. Herein, we apply the modified equation technique in an original way, by switching the classical discretization process. Indeed, we consider first the time discretization, thanks to the modified equation technique, before addressing the question of the space one. After this time discretization, we have to deal with an additional p-laplacian operator which implies to consider $C^{p-1}$ finite elements. In this work, we have chosen to discretize the second-order operator by the Interior Penalty Discontinuous Galerkin method and we will present how we extend this method to discretize the higher-order operators. Numerical results in 1D and 2D illustrate the performance of the 4-th and 6-th order schemes and show that our technique can be extended to deal with elements of various order in the same mesh.
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Cyril Agut, Julien Diaz, Abdelaâziz Ezziani. High-order discretizations for the wave equation based on the modified equation technique. 10ème Congrès Français d'Acoustique, Apr 2010, Lyon, France. ⟨hal-00554448⟩

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