A Nearly Analytic Exponential Time Difference Method For Solving Wave Equations And Its Numerical Simulation

The forward modeling for seismic wave propagation is the hotspot in the fields ofGeophysics and petroleum exploration. In this study, we propose a new type ofnumerical method for solving wave equations based on the structure of the Lie groupmethod by combing the exponential time difference (ETD) method and the nearlyanalytic discrete (NAD) operator, which is called nearly analytic exponential timedifference (NETD) method. We first introduce the particle velocity and the gradientfield into the wave equation and transform the second-order wave equation intofirst-order partial differential equations. Then on the base of the idea of the nearlyanalytic discrete method, we use the linear combination of displacement, particlevelocity and their gradients to approximate the high-order spatial derivatives andconvert the first-order differential equations into semi-discrete ordinary differentialequations (ODEs). Finally, we use the exponential time difference (ETD) method tosolve the converted ODEs. To improve the stability of the scheme, we first reform thescheme into an implicit scheme and then explicitly solve the linear equations to obtainan explicit scheme.In this paper, we investigate the NETD method in detail from theoretical analysesand numerical experiments. The main contents are listed as follows. Firstly, weinvestigate the stability of the NETD method for solving the1-D and2-D acoustic waveequations. Secondly, we derive the theoretical error of the NETD method. We show thenumerical error of the NETD method for1-D and2-D initial problems and compare itwith the Lax-Wendroff correction method and staggered grid method. Thirdly, weinvestigate the numerical dispersion of the NETD method for1-D and2-D acousticwave equations and compare dispersion curves of the NETD method with those of theLax-Wendroff correction method and staggered grid method. Fourthly, we compare thecomputational efficiency of the NETD method with those of the Lax-Wendroffcorrection method and staggered grid method. Finally, to examine the performance ofthe NETD method in the numerical simulation, we apply the NETD method to simulatethe acoustic/elastic wave propagation in multilayer and heterogeneous media.The theoretical analyses show that the NETD method is based on the Lie groupmethod, and it constructs the numerical solution from the point of view of the one-parameter group of transformations and flow. This means the NETD methoddescribes the movement of the particle in the elastomers more exactly. Thus itcansimulate the seismic wave propagation more accurately. On the other hand, theNETD method uses the nearly analytic discrete operator to approximate the high-orderspatial derivatives, which means that the spatial discretization contains moreinformation of the wave field. Thus the NETD method has small dispersion error. Thenumerical results show that the NETD method has small relative error and caneffectively suppress the numerical dispersion when coarse grid is used or velocity modelhas strong contrasts cross the interface. This means the NETD method can improve thecomputational speed and reduce the storage by using large spatial and temporalincrements. In a word, the NETD method has great potentiality of applying inGeophysics and petroleum exploration.