scholarly journals Space-time spectral element method solution for the acoustic wave equation and its dispersion analysis

2017 ◽  
Vol 38 (6) ◽  
pp. 303-313 ◽  
Author(s):  
Yanhui Geng ◽  
Guoliang Qin ◽  
Jiazhong Zhang ◽  
Wenqiang He ◽  
Zhenzhong Bao ◽  
...  
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Spectral Element Method ◽  
Spectral Element ◽  
Dispersion Analysis ◽  
Space Time ◽  
Acoustic Wave Equation ◽  
Element Method

A Space-Time Discontinuous Galerkin Spectral Element Method for Nonlinear Hyperbolic Problems

2018 ◽  
Vol 16 (01) ◽  
pp. 1850093 ◽  
Author(s):  
Chaoxu Pei ◽  
Mark Sussman ◽  
M. Yousuff Hussaini
Keyword(s):  
Discontinuous Galerkin ◽  
Spectral Element Method ◽  
Burgers Equation ◽  
Spectral Element ◽  
Space Time ◽  
Picard Iteration ◽  
Spectral Accuracy ◽  
Element Discretization ◽  
Space And Time ◽  
Element Method

A space-time discontinuous Galerkin spectral element method is combined with two different approaches for treating problems with discontinuous solutions: (i) adding a space-time dependent artificial viscosity, and (ii) tracking the discontinuity with space-time spectral accuracy. A Picard iteration method is employed to solve nonlinear system of equations derived from the space-time DG spectral element discretization. Spectral accuracy in both space and time is demonstrated for the Burgers’ equation with a smooth solution. For tests with discontinuities, the present space-time method enables better accuracy at capturing the shock strength in the element containing shock when higher order polynomials in both space and time are used. The spectral accuracy of the shock speed and location is demonstrated for the solution of the inviscid Burgers’ equation obtained by the tracking method.

scholarly journals Efficiency of the spectral element method with very high polynomial degree to solve the elastic wave equation

Geophysics ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. T33-T43
Author(s):  
Chao Lyu ◽  
Yann Capdeville ◽  
Liang Zhao
Keyword(s):  
Wave Equation ◽  
Spectral Element Method ◽  
Computing Time ◽  
Spectral Element ◽  
Two Dimensions ◽  
Elastic Model ◽  
Time Step ◽  
High Degree ◽  
Very High ◽  
Element Method

The spectral element method (SEM) has gained tremendous popularity within the seismological community to solve the wave equation at all scales. Classic SEM applications mostly rely on degrees 4–8 elements in each tensorial direction. Higher degrees are usually not considered due to two main reasons. First, high degrees imply large elements, which make the meshing of mechanical discontinuities difficult. Second, the SEM’s collocation points cluster toward the edge of the elements with the degree, degrading the time-marching stability criteria and imposing a small time step and a high numerical cost. Recently, the homogenization method has been introduced in seismology. This method can be seen as a preprocessing step before solving the wave equation that smooths out the internal mechanical discontinuities of the elastic model. It releases the meshing constraint and makes use of very high degree elements more attractive. Thus, we address the question of memory and computing time efficiency of very high degree elements in SEM, up to degree 40. Numerical analyses reveal that, for a fixed accuracy, very high degree elements require less computer memory than low-degree elements. With minimum sampling points per minimum wavelength of 2.5, the memory needed for a degree 20 is about a quarter that of the one necessary for a degree 4 in two dimensions and about one-eighth in three dimensions. Moreover, for the SEM codes tested in this work, the computation time with degrees 12–24 can be up to twice faster than the classic degree 4. This makes SEM with very high degrees attractive and competitive for solving the wave equation in many situations.

Dispersion analysis of the spectral element method

2012 ◽  
Vol 138 (668) ◽  
pp. 1934-1947 ◽  
Author(s):  
Thomas Melvin ◽  
Andrew Staniforth ◽  
John Thuburn
Keyword(s):  
Spectral Element Method ◽  
Spectral Element ◽  
Dispersion Analysis ◽  
Element Method
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