In this study, we develop a 2d finite difference solver for the elastic wave equation that combines a 4th order ader scheme arbitrary highorder accuracy using derivatives, which is widely used in aeroacoustics, with the characteristic variable method at the free surface boundary. We verify the order of the method for smooth materials and its discrete con. Secondorder accurate explicit schemes laxwendroff,upwind schemes give excellent results with a min of computational effort. A high order compact timespace finite difference scheme. Optimal staggeredgrid finitedifference schemes based on. Higher order compact implicit schemes for the wave equation. For this we investigate finite difference method and present explicit upwind difference scheme for one dimensional wave equation, central difference scheme for. In this paper, we present a stable and consistent criterion to an explicit finite difference scheme for a timedependent schrodinger wave equation. Assuming that thewavefieldvalues of several neighbor. Compared with conventional standardgrid finite difference fd methods, staggeredgrid fd sgfd methods have been more widely used in numerically solving wave equations for their greater precision and stability, though some nonstaggered schemes e. Finite difference methods for hyperbolic equations. Finite difference methods for solving differential equations iliang chern department of mathematics national taiwan university may 16, 20. Schemes for the twodimensional wave equation brian hamilton. An efficient and high accuracy finitedifference scheme for.
Thus, compact schemes results in matrices that have smaller bandwidth compared with noncompact schemes that involves more grid points. Nov 28, 2020 arxivlabs is a framework that allows collaborators to develop and share new arxiv features directly on our website. The derivation can be found in mitchell 1980, p198199. Finite difference methods for differential equations edisciplinas. These schemes will be linear and shiftinvariant, and as such, it. Secondorder stable finite difference schemes for the time. For the wave equation, a finite difference scheme with fourth order accuracy in. 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. Later, we use this observation to conclude that bordings conjecture for stability of finite difference schemes for the scalar wave equation lines et al. This paper investigates some fourthorder accurate explicit finite difference schemes for the 2d wave equation obtained using, 17, 21, and 25point discrete laplacians. The finite difference method fdm is used for transformation of wave. Some numerical and analytical methods for equations of wave.
Numerical experiments in one and twospace dimensions illustrating the stability and. In this article we discuss the numerical analysis for the finite difference scheme of the onedimensional nonlinear wave equations with dynamic boundary conditions. A neutrally stable scheme is one in which errors remain constant as the computations are carried forward. Finite difference schemes for the wave equation edinburgh. Feb 01, 2020 in this paper we developed a new explicit compact highorder finite difference scheme to solve the 3d acoustic wave equations with spatially variable acoustic velocity. 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 results for different particular cases of.
A formal derivation of the method from the boltzmann equation with grazing. An efficient and high accuracy finitedifference scheme. Finite difference method for the diffusiveviscous wave equation haixia zhao, jinghuai gao, and zhangxin chen abstract. Solving wave equation using finite differences and. Pdf a fully implicit finite difference scheme for the. In the equations of motion, the term describing the transport process is often called convection or advection. We begin our study of wave equations by simulating onedimensional waves on a string, say on a guitar or violin.
The computer program is also developed in lahey ed developer and for graphical representation tecplot 7 software is. Fourth order symmetric finite difference schemes for the. The technique is illustrated using excel spreadsheets. The results suggest that 84000 nodes could be accommodated on a single virtex ii fpga. Fuzzy numerical solution via finite difference scheme of wave. A new six point finite difference scheme, which is equivalent to the multisymplectic integrator, is derived for the model equations. Finite difference modelling of the full acoustic wave. We begin our study of wave equations by simulating onedimensional waves on a. Finite difference scheme for the zakharov equation as a. The center is called the master grid point, where the finite difference equation is used to approximate the pde. Finite difference method has been used to solve a variety of physical prob lems, 9, 16, 29. As a consequence, the period of the wave is slightly di. Finite difference schemes for the wave equation 3 figure 1. Numerical analysis lecture notes math user home pages.
Comparison of finite difference schemes for the wave. Finite difference fd schemes transform the partial differential equation into a difference equation. Convergence rates of finite difference schemes for the. Difference schemes for the wave equation abraham zemui uppsala university department of information technology. Method, the heat equation, the wave equation, laplaces equation. A stable and consistent finite difference scheme for a time. Finitedifference schemes for nonlinear wave equation that. Optimisation is conducted in order to minimise numerical dispersion and computational costs.
Robust finite difference schemes for a nonlinear variational. Finite difference schemes for the wave equation in this appendix, we reexamine the. A finite difference scheme is stable if the errors made at one time step of the calculation do not cause the errors to be magnified as the computations are continued. The study of schemes with conservation property was initiated by courant et al. This paper presents a secondorder numerical scheme, based on nite di erences, for solving the wave equation in polar and cylindrical domains. Finite differences for the wave equation ucsb math. In section 6, two numerical examples are presented. The numerical simulations are also presented for the model. See standard pde books such as kev90 for a derivation and more. Exact finite difference schemes for solving helmholtz equation 93 continuous problem. For the linear wave equation, there is no major difference between the drp and opc schemes.
Both individuals and organizations that work with arxivlabs have embraced and accepted our values of openness, community, excellence, and user data privacy. Pdf in the present paper a numerical solution of the regularized long wave rlw equation with a fully implicit finite difference method is deduced find, read and cite all the research you. Finite difference approximations to derivatives, the finite difference method, the heat equation. A stable and consistent finite difference scheme for a. On compact 4th order finitedifference schemes for the wave.
Higher order finite difference discretization for the wave equation the two dimensional version of the wave equation with velocity and acoustic pressure v in homogeneous mu edia can be written as 2 22 2 2 22, u uu v t xy. Wave equation in double parametrical fuzzy number form. The last equation is a finitedifference equation, and solving this equation gives an approximate solution to the differential equation. Timedomain numerical solution of the wave equation. Introductory finite difference methods for pdes contents contents preface 9 1. The paper explores comparably low dispersive scheme with among the finite difference schemes. Finite difference scheme for the zakharov equation as a model. In figure 3 we show the error as function of normalized time, tt, for the two materials. Dec 12, 2014 we propose two stable and one conditionally stable finite difference schemes of secondorder in both time and space for the timefractional diffusion wave equation. Finite difference methods for wave motion hans petter. More complicated shapes of the spatial domain require substantially more advanced techniques and implementational efforts and a finite element method is usually a more.
Numerical simulation of wave equation global journals. From the viewpoint of the discrete variational derivative method we propose the derivation of the structurepreserving finite difference schemes of the. Finite difference schemes for the wave equation abraham zemui dec. Solve 1d wave equation using finite difference method. Numerical solution of partial differential equations uq espace. A secondorder finite di erence scheme for the wave equation on a reduced polar grid abstract. We consider a nonlinear variational wave equation that models the dynamics of nematic liquid crystals. Nonstandard finite difference scheme for helmholtz wave equation marek moszynski gdansk university of technology, ul. Construction analysis of fourthorder finite difference. Pdf numerical source implementation in a 2d finite. Equation 8 suggests that the finite difference scheme for the divergence is of the same secondorder form. Finite difference scheme, wave wave interaction, 1. Finite difference methods for the hyperbolic wave pde. For first order hyperbolic equations, technical assumptions of finite difference schemes are.
We do this for a particular case of a finitely low potential well. In the present paper a numerical solution of the regularized long wave rlw equation with a fully implicit finite difference method is deduced. Finite difference methods for 2d and 3d wave equations. Similar to the numerical schemes for the heat equation, we can use approximation of derivatives by difference quotients to arrive at a numerical scheme for the. New schemes are presented that are more computationally efficient than ninepoint explicit schemes at maintaining less than one. Comparison of finite difference schemes for the wave equation. A general finite difference scheme the 2d scalar wave constantdensity acoustic wave equation in the frequency domain is given by.
The proof of uniqueness for the discrete problems are presented. Finite difference methods for hyperbolic equations 1. Explicit finite difference schemes for first order 1d wave equation. Finite difference elastic wave modeling with an irregular. 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. On compact 4th order finitedifference schemes for the. For the nonlinear wave equations, the finite volume version of both drp and opc. Solution of third order viscous wave equation using finite difference. Solving the heat, laplace and wave equations using. The finite difference equation at the grid point involves five grid points in a fivepoint stencil. Higher order finite difference techniques have associated large star systems which engender complications near the boundary. A secondorder finite di erence scheme for the wave. In the first scheme, we apply the fractional trapezoidal rule in time and the central difference in space. This socalled energy method attracted widespread attention in 1950s, as is documented by richtmyer and morton 27, section 6.
This paper is a departure from the wellestablished time independent schrodinger wave equation swe. This section shows derivation of npoint difference formula using taylor series. Finally, we conclude the paper with a short remark in section 4. Solutions of kinematic wave equations through finite difference method crank nicolson and finite element method are developed for this study. A high order compact timespace finite difference scheme for. Introduction physical modelling based sound synthesis and acoustical simulation deals with the solution of partial differential equations representing physical phenomena of sound. If the errors decay and eventually damp out, the numerical scheme is said. 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. A general optimal method for a 2d frequencydomain finite. The solution of the acoustic wave equation in one space dimension is studied. Finite difference approximations to onedimensional. Anders petersson may 11, 2011, revised aug 8, 2011 published in journal of scientific computing, v.
Pdf implementation of finite difference schemes for the. In section 4 we give several numerical examples in two and three dimensions. Higher order compact implicit schemes for the wave equation by melvyn ciment and stephen h. Pdf finite difference techniques are widely used for the numerical simulation of timedependent partial differential equations. 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. A stable finite difference method for the elastic wave equation on. Since both time and space derivatives are of second order, we use centered di erences to approximate them. A finite difference method proceeds by replacing the derivatives in the differential equations.
Introduction most hyperbolic problems involve the transport of fluid properties. May 21, 2004 summary this appendix contains sections titled. A secondorder finite di erence scheme for the wave equation. Rather than discretizing second order hyperbolic equations such as the wave. Llnljrnl483427 l aw r e n c e livermore n at i o n a l a fourth order accurate finite laboratory difference scheme for the elastic wave equation in second order formulation bjorn sjogreen and n. Finite difference method for solving differential equations. In particular, one is interested in designing numerical methods that can be rigorously. A finite difference equation formulation for the equations of elasticity is presented and applied to the problem of a layered halfspace with a buried point source emitting a compressional pulse. Convergence of finite difference methods for the wave equation in. Understand what the finite difference method is and how to use it to solve problems. Nonstandard finite difference scheme for helmholtz wave equation proceedings of the 38th scandinavian symposium on physical acoustics, geilo 30 february 2 6, 20. The two dimensional version of the wave equation with velocity and acoustic pressure v in homogeneous mu e dia can be written as. Two finite difference schemes for time fractional diffusion.
Pdf comparison of finite difference schemes for the wave. Sep 01, 2001 we consider finite difference schemes that inherit energy conservation property from nonlinear wave equations. The boundary conditions for the second derivatives of spatial variables have been derived by using the wave equation and the boundary conditions themselves. Consider the normalized heat equation in one dimension, with homogeneous dirichlet boundary conditions. Finite difference method applied to 1d convection in this example, we solve the 1d convection equation. Numerical simulations for onedimensional and twodimensional problems are reported in section 3. Implicit scheme is probably not the optimum choice.
149 301 1482 1205 1688 531 746 167 712 1261 896 177 1332 532 1214 308 1481 120 1076 1351 1561 230 38 1135 545 1262 1295 899