In this paper, we consider the numerical approximation of the threedimensional poroelastic wave equations in the spherical coordinate system. We study stability and consistency of these methods. Feb 15, 2011 in this paper, a class of finite difference method for solving twosided spacefractional wave equation is considered. Nov 12, 2020 the finite difference method, by applying the threepoint central difference approximation for the time and space discretization. The finite difference method for the helmholtz equation with.
Finite difference method for solving differential equations. Finite elementsstatic and timedependent pdesseismic wave propagation geophysical. Finite difference methods for the oneway wave equation author. When using a finite difference method to solve an initialboundary value problem, the truncation error is often of lower order at a few. Introduction most hyperbolic problems involve the transport of fluid properties.
Pdf comparison of finite difference schemes for the wave. We shall now describe in detail various python implementations for solving a standard 2d, linear wave equation with constant wave velocity and \u0\ on the boundary. Many types of wave motion can be described by the equation u tt. Sep 01, 2020 in this paper, a completely new and pure semi finite difference scheme for a 1d wave equation with local viscosity damping is proposed by a semidiscrete finite difference scheme. Elastic wave equation, curvilinear grids, finite differences, stability, energy estimate, seismic wave propagation. Introductory finite difference methods for pdes contents contents preface 9 1.
Finite difference methods massachusetts institute of. Finite difference methods for the hyperbolic wave pde. Comparison of finite difference schemes for the wave equation. Introductory finite difference methods for pdes department of. Sen abstract the staggeredgrid finite difference sfd method is widely used in numerical modeling of wave equations.
The discretization of the spatial operators in the. Step wave test for the lax method to solve the advection % equation clear. Scalar wave equation modeling with timespace domain dispersionrelationbased staggeredgrid finitedifference schemes by yang liu and mrinal k. Solving wave equation using finite differences and. The accuracy will be obtained by comparing the value of finite difference method with. Numerical solution of fractional diffusionwave equation. The acoustic wave equation for a twodimensional 2d problem, defined in the. An obvious extension is to incorporate variable density.
Jan 24, 20 time fractional diffusion wave equations are generalizations of classical diffusion and wave equations which are used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. Finite differences for the wave equation ucsb math. Elliptic equations and errors, stability, lax equivalence theorem. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. Solving the heat, laplace and wave equations using. Finite difference method for hyperbolic problems free download as powerpoint presentation. The paper deals with the numerical solution of partial differential equations pdes, especially wave equation. Standard finite difference methods for the scalar wave equation have been implemented as part of the crewes matlab toolbox by youzwishen and margrave 1999 and margrave 2000. A stable and explicit second order accurate finite difference method for the elastic wave equation in curvilinear coordinates is presented. The ctcs method is the classic explicit scheme to approach the wave equation which can be very accurate under certain restriction. Numerical modeling of wave equation by a trun cated high. A secondorder finite di erence scheme for the wave equation. It can be advantageous to modify the coefficients of a potentially higherorder method, thereby.
Similarly, higherorder finite difference methods have been developed for acoustic applications by gottlieb and turkel 22, cohen and joly 23, and davis 24. Finite difference modelling of the full acoustic wave. Scalar wave equation modeling with time space domain. Explicit finite difference schemes for first order 1d wave equation. Solve 1d wave equation using finite difference method. In particular, the left term in equation 5 says that the derivative of the e field at time n. First, the wave equation is presented and its qualities analyzed. The wave equation considered here is an extremely simplified model of the physics of waves. Equations 5 and 6 show the usefulness of yees scheme in order to have a central difference approximation for the derivatives.
Numerical methods for maxwells equations summer term 2014. We develop an energybased finite difference method for the wave equation in second order form. It has been applied to solve a time relay 2d wave equation. Taking the fourier transform of this recursion gives. Finite difference methods for the oneway wave equation. More precisely, the complete initialboundary value problem is defined by. In this article, a novel compact finite difference scheme is \mboxconstructed to solve the fractional diffusion wave equation based on its equivalent integrodifferential equation. A natural next step is to consider extensions of the methods for various variants of the onedimensional wave equation to twodimensional 2d and threedimensional 3d versions of the wave equation. Finite volumestimedependent pdesseismic wave propagation geophysical fluid dynamics maxwells equations ground penetrating radar robust, simple concept, easy to. Fast fourier transform guest lecture by steven johnson 9. Pdf a stable finite difference method for the elastic wave. The finite difference equation at the grid point involves five grid points in a fivepoint stencil.
Later, we use this observation to conclude that bordings conjecture for stability of finite difference schemes for the scalar wave equation lines et al. The first order wave equation in onedimensional space is as follows. A stable finite difference method for the elastic wave equation on. Solve 1d wave equation using finite difference method youtube. The spatial discretization satisfies a summationbyparts sbp property.
Finite difference methods are a prevalent computational technique that applies to variable coefficient as well as nonlinear problems. Solutions of kinematic wave equations through finite difference method crank nicolson and finite element method are developed for this study. The finite difference method is used to solve ordinary differential equations that have. Finite difference and discontinuous galerkin methods for wave. One difficulty in the design of an efficient numerical scheme is that the problem is singular in the center and the polar axes of the computational domain. Explicit finite difference method is often used to find numerical solution of this.
We do not add any artificial numerical viscosity term as most of the existing literature in the discrete scheme. The di usiveviscous wave equation plays an important role in seismic exploration and it can be used to explain the frequencydependent re ections observed both in laboratory and eld data. Numerical solutions of some wave fractional partial differential equation models are presented. Finite difference methods for hyperbolic equations. The method was introduced by runge in 1908 to understand the torsion in a beam of arbitrary cross section, which results in having to solve a poisson equation. Comparison of finite difference schemes for the wave. A semidiscrete finite difference method to uniform. Convergence of finite difference methods for the wave equation in. Finite difference method for the diffusiveviscous wave equation haixia zhao, jinghuai gao, and zhangxin chen abstract.
Two methods are explicit and they approximates the solutions of the wave equation with consistency of. A secondorder finite di erence scheme for the wave. The two dimensional version of the wave equation with velocity and acoustic pressure v in homogeneous mu e dia can be written as. Lecture notes numerical methods for partial differential. Numerical solution of twosided spacefractional wave. A general framework for deriving higherorder finite difference schemes was proposed by lynch and rice for ordinary differential equations 1 and elliptic partial differential equations 2, and applied to the. Finite difference methods for wave motion hans petter. We apply the finite difference method to determine numerical solutions.
This paper presents a secondorder numerical scheme, based on nite di erences, for solving the wave equation in polar and cylindrical domains. Two finite difference schemes for time fractional diffusion. Indeed, the lessons learned in the design of numerical algorithms for solved examples are of inestimable. Finite difference method applied to 1d convection in this example, we solve the 1d convection equation.
Pdf finite difference method for solving acoustic wave equation. Pdf a stable finite difference method for the elastic. The center is called the master grid point, where the finite difference equation is used to approximate the pde. Pdf numerical solution of fractional wave equation using. Solution of differential equation by finite difference method. In the equations of motion, the term describing the transport process is often called convection or advection.
These implementations handle a variablevelocity subsurface and a variety of simple boundary conditions. Numerical solution of partial differential equations uq espace. Finite difference method for solving acoustic wave equation using. The 1d wave equation hyperbolic prototype the 1dimensional wave equation is given by. Pdf finite difference techniques are widely used for the numerical simulation of timedependent partial differential equations. A hybrid finite differencecontrol volume method for the. Conventional sfd stencils for spatial derivatives are usually designed in the. The accuracy will be obtained by comparing the value of finite difference method with the analytical solution. The straightforward finite difference approximation to the second partial derivative is. Finite difference methods for hyperbolic equations 1. Sen 2 1 state key laboratory of petroleum resource and prospecting, china university of petroleum, beijing 102249, china 2 institute for geophysics, john a. The stability and consistency of the method are discussed by means of gerschgorin theorem and using the stability matrix analysis. The finite difference method is an easytounderstand method for obtaining approximate solutions of pdes. A secondorder finite di erence scheme for the wave equation on a reduced polar grid abstract.
Partial differential equations draft analysis locally linearizes the equations if they are not linear and then separates the temporal and spatial dependence section 4. So, we will take the semidiscrete equation 110 as our starting point. In this section, we solved the damped wave equation using finite difference method at time step, where the value of k is, while at step size, and. Higher order compact finite difference method for the wave equation a compact finite difference scheme comprises of adjacent point stencils of which differences are taken at the middle node, therefore typically 3, 9 and 27 nodes are used for compact finite difference descretization in one.
Nevertheless, we develop a hybrid finite difference control volume method for solving this problem. Central time central space ctcs, cranknicolson and. The computer program is also developed in lahey ed developer and for graphical representation tecplot 7 software is. Understand what the finite difference method is and how to use it to solve problems. Pdf comparison of two finite difference methods for solving the. In this paper we construct two finite difference schemes to solve a class of initialboundary value time fractional diffusion wave equations based on its equivalent partial integro.
With boundary conditions and material interface conditions imposed weakly by the simultaneousapproximationterm sat method, we derive energy estimates for the semidiscretization. Pdf explicit finite difference method has been widely used for seismic modeling in heterogeneous media with strong discontinuities in. These three equations are known as the prototype equations, since many homogeneous linear second order pdes in. Some numerical and analytical methods for equations of wave. Pdf an energybased summationbyparts finite difference. General finite difference approach and poisson equation. A computational study with finite difference methods for. Bancroft abstract a new method of migration using the finite element method fem and the finite difference method fdm is jointly used in the spatial domain. Finite difference methods for 2d and 3d wave equations. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous. With boundary conditions and material interface conditions imposed weakly by the simultaneousapproximationterm sat method, we derive energy estimates for the semi. The wave equation describes how the sound pressure p propagates at speed c 0 through a medium at rest, and plays a role in acoustics, fluid mechanics, and quantum mechanics. Before doing that, we need to discretize the space of solutions.
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