Conventional finite-difference(FD)methods cannot model acoustic wave propagation beyond Courant-Friedrichs-Lewy(CFL)numbers 0.707 and 0.577 for two-dimensional(2D)and three-dimensional(3D)equal spacing cases,respectiv...Conventional finite-difference(FD)methods cannot model acoustic wave propagation beyond Courant-Friedrichs-Lewy(CFL)numbers 0.707 and 0.577 for two-dimensional(2D)and three-dimensional(3D)equal spacing cases,respectively,thereby limiting time step selection.Based on the definition of temporal and spatial FD operators,we propose a variable-length temporal and spatial operator strategy to model wave propagation beyond those CFL numbers while preserving accuracy.First,to simulate wave propagation beyond the conventional CFL stability limit,the lengths of the temporal operators are modified to exceed the lengths of the spatial operators for high-velocity zones.Second,to preserve the modeling accuracy,the velocity-dependent lengths of the temporal and spatial operators are adaptively varied.The maximum CFL numbers for the proposed method can reach 1.25 and 1.0 in high velocity contrast 2D and 3D simulation examples,respectively.We demonstrate the effectiveness of our method by modeling wave propagation in simple and complex media.展开更多
This paper is concerned with the weak Galerkin mixed finite element method(WG-MFEM)for the second-order hyperbolic acoustic wave equation in velocity-pressure formulation.In this formulation,the original second-order ...This paper is concerned with the weak Galerkin mixed finite element method(WG-MFEM)for the second-order hyperbolic acoustic wave equation in velocity-pressure formulation.In this formulation,the original second-order differential equation in time and space is reduced to first-order differential equations by introducing the velocity and pressure variables.We employ the usual discontinuous piecewise-polynomials of degree k0 for the pressure and k+1 for the velocity.Furthermore,the normal component of the pressure on the interface of elements is enhanced by polynomials of degree k+1.The time derivative is approximated by the backward Euler difference.We show the stability of the semi-discrete and fullydiscrete schemes,and obtain the suboptimal order error estimates for the velocity and pressure variables.Numerical experiment confirms our theoretical analysis.展开更多
We present a new high ordermethod in space and time for solving the wave equation,based on a newinterpretation of the“Modified Equation”technique.Indeed,contrary to most of the works,we consider the time discretizat...We present a new high ordermethod in space and time for solving the wave equation,based on a newinterpretation of the“Modified Equation”technique.Indeed,contrary to most of the works,we consider the time discretization before the space discretization.After the time discretization,an additional biharmonic operator appears,which can not be discretized by classical finite elements.We propose a new Discontinuous Galerkinmethod for the discretization of this operator,andwe provide numerical experiments proving that the new method is more accurate than the classicalModified Equation technique with a lower computational burden.展开更多
Finite difference method is an important methodology in the approximation of waves.In this paper,we will study two implicit finite difference schemes for the simulation of waves.They are the weighted alternating direc...Finite difference method is an important methodology in the approximation of waves.In this paper,we will study two implicit finite difference schemes for the simulation of waves.They are the weighted alternating direction implicit(ADI)scheme and the locally one-dimensional(LOD)scheme.The approximation errors,stability conditions,and dispersion relations for both schemes are investigated.Our analysis shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme.Moreover,the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time.In order to improve computational efficiency,numerical algorithms based on message passing interface(MPI)are implemented.Numerical examples of wave propagation in a three-layer model and a standard complex model are presented.Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.展开更多
基金the National Natural Science Foundation of China(No.41874144)the Research Foundation of China University of PetroleumBeijing at Karamay(RCYJ2018A-01-001).
文摘Conventional finite-difference(FD)methods cannot model acoustic wave propagation beyond Courant-Friedrichs-Lewy(CFL)numbers 0.707 and 0.577 for two-dimensional(2D)and three-dimensional(3D)equal spacing cases,respectively,thereby limiting time step selection.Based on the definition of temporal and spatial FD operators,we propose a variable-length temporal and spatial operator strategy to model wave propagation beyond those CFL numbers while preserving accuracy.First,to simulate wave propagation beyond the conventional CFL stability limit,the lengths of the temporal operators are modified to exceed the lengths of the spatial operators for high-velocity zones.Second,to preserve the modeling accuracy,the velocity-dependent lengths of the temporal and spatial operators are adaptively varied.The maximum CFL numbers for the proposed method can reach 1.25 and 1.0 in high velocity contrast 2D and 3D simulation examples,respectively.We demonstrate the effectiveness of our method by modeling wave propagation in simple and complex media.
基金supported by the National Natural Science Foundation of China(No.11971337)the Key Fund Project of Sichuan Provincial Department of Education(No.18ZA0276).
文摘This paper is concerned with the weak Galerkin mixed finite element method(WG-MFEM)for the second-order hyperbolic acoustic wave equation in velocity-pressure formulation.In this formulation,the original second-order differential equation in time and space is reduced to first-order differential equations by introducing the velocity and pressure variables.We employ the usual discontinuous piecewise-polynomials of degree k0 for the pressure and k+1 for the velocity.Furthermore,the normal component of the pressure on the interface of elements is enhanced by polynomials of degree k+1.The time derivative is approximated by the backward Euler difference.We show the stability of the semi-discrete and fullydiscrete schemes,and obtain the suboptimal order error estimates for the velocity and pressure variables.Numerical experiment confirms our theoretical analysis.
基金supported by the Conseil General des Pyrenees Atlantiques.
文摘We present a new high ordermethod in space and time for solving the wave equation,based on a newinterpretation of the“Modified Equation”technique.Indeed,contrary to most of the works,we consider the time discretization before the space discretization.After the time discretization,an additional biharmonic operator appears,which can not be discretized by classical finite elements.We propose a new Discontinuous Galerkinmethod for the discretization of this operator,andwe provide numerical experiments proving that the new method is more accurate than the classicalModified Equation technique with a lower computational burden.
基金supported by the State Key Project(2010CB731505)National Key Foundation Project(10431030)Director Foundation Project of Laboratory of Scientific and Engineering Computing(LSEC).
文摘Finite difference method is an important methodology in the approximation of waves.In this paper,we will study two implicit finite difference schemes for the simulation of waves.They are the weighted alternating direction implicit(ADI)scheme and the locally one-dimensional(LOD)scheme.The approximation errors,stability conditions,and dispersion relations for both schemes are investigated.Our analysis shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme.Moreover,the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time.In order to improve computational efficiency,numerical algorithms based on message passing interface(MPI)are implemented.Numerical examples of wave propagation in a three-layer model and a standard complex model are presented.Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.