Time exponential splitting technique for the Klein–Gordon equation with Hagstrom–Warburton high-order absorbing boundary conditions

Journal of Computational and Applied Mathematics

2018

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The Klein-Gordon equation on an infinite two dimensional strip is considered. Numerical computation is reduced to a finite domain by using the Hagstrom-Warburton (H-W) absorbing boundary conditions (ABCs) with free parameters in the formulation of the auxiliary variables. The spatial discretization is achieved by using fourth order finite differences and the time integration is made by means of an efficient and easy to implement fourth order exponential splitting scheme which was used in [5] considering the fixed Padé parameters in the formulation of the ABCs. Here, we generalize the splitting time technique to other choices of the parameters. To check the time integrator we consider, on one hand, four types of fixed parameters, the Newmann's parameters, the Chebyshev's parameters, the Padé's parameters and optimal parameters proposed in [13] and, on the other hand, an adaptive scheme for the dynamic control of the order of absorption and the parameters. We study the efficiency of the splitting scheme by comparing with the fourth-order four-stage Runge-Kutta method.

Section: Time Integrationmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

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Time exponential splitting integrator for the Klein–Gordon equation with free parameters in the Hagstrom–Warburton absorbing boundary conditions

Alonso‐Mallo

Journal of Computational and Applied Mathematics

2018

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“…Following a reasoning similar to the one done in [15,20], the eigenvalues of k(−A) 1/2 must be in the stability interval…”

Section: Stability Discussionmentioning

confidence: 99%

“…Although in the current paper we have focused on periodic boundary conditions, other interesting problems may need to consider artificial boundary conditions. We leave for future work to study if the proposed methods can handle for example absorbing boundary conditions, as we did in [15] with similar splitting methods, for the Klein-Gordon equation with Hagstrom-Warburton absorbing boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Near conserving energy numerical schemes for two-dimensional coupled seismic wave equations

Computers & Mathematics with Applications

2018

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Two-dimensional coupled seismic waves, satisfying the equations of linear isotropic elasticity, on a rectangular domain with initial conditions and periodic boundary conditions, are considered. A quantity conserved by the solution of the continuous problem is used to check the numerical solution of the problem. Second order spatial derivatives, in the x direction, in the y direction and mixed derivative, are approximated by finite differences on a uniform grid. The ordinary second order in time system obtained is transformed into a first order in time system in the displacement and velocity vectors. For the time integration of this system, second order and fourth order exponential splitting methods, which are geometric integrators, are proposed. These explicit splitting methods are not unconditionally stable and the stability condition for time step and space step ratio is deduced. Numerical experiments displaying the good behavior in the long time integration and the efficiency of the numerical solution are provided.

High-order full discretization for anisotropic wave equations

Applied Mathematics and Computation

2018

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