<i>Difference Methods for Initial-Value Problems</i>

Richtmyer, Robert D.; Morton, K. W.

doi:10.1063/1.3060778

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“…However, we had no difficulty in inverting the matrices Fj. An analysis by the method of von Neumann [4] shows this scheme to be unconditionally stable and this conclusion is supported by numerical experiments described in the following section.…”

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On certain finite difference schemes for hyperbolic systems

Gary¹

1964

Math. Comp.

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1. Introduction. The purpose of this paper is to describe a few finite difference schemes for the numerical solution of hyperbolic systems of partial differential equations. Our main concern will be the stability conditions for these schemes although some of these schemes may be useful, especially for problems involving supersonic fluid flow. We will describe numerical experiments designed to check the derived stability conditions and also the accuracy of these schemes.In Section 2 we will describe the application of the Crank-Nicholson scheme to hyperbolic systems. This method is unconditionally stable, however it requires the inversion of a block tridiagonal matrix. We describe two modifications which require only the inversion of a scalar tridiagonal matrix. We carry out a stability analysis which shows that these schemes are unconditionally stable only if the matrix of the system is positive definite (supersonic flow in the case of hydrodynamics). Results of numerical computations using all these schemes are described in Section 3. All our results are for problems in one space dimension. These methods can be generalized to two dimensions but we have no analysis to indicate that the generalizations will work. In Section 5 we give the results of fluid dynamics computations using three versions of the Lax-Wendroff difference scheme. The objective here is to determine if it is necessary to write the equations in conservation form in order to obtain good results when the flow contains a shock. In Section 6 an application of the Lax-Wendroff scheme to the Navier-Stokes equations is described. An empirical stability condition is obtained which is a combination of the usual hyperbolic and parabolic conditions. In Section 7 an explicit difference scheme related to the CrankNicholson scheme is given. This is an iterative scheme with a rather peculiar sta-

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“…In this section we will describe some finite difference schemes of an implicit nature. For purposes of comparison we will begin with the usual Crank-Nicholson scheme [4]. This scheme is adapted to the nonlinear problem by use of a "predictor-corrector" method.…”

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On certain finite difference schemes for hyperbolic systems

Gary¹

1964

Math. Comp.

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“…Notable examples are the dissipative Lax-Friedrichs and Lax-Wendroff schemes, the unitary Leap-Frog scheme and the implicit backward Euler and Crank-Nicolson schemes [175,95]. The computation of multidimensional probelms is often accomplished using splitting methods such as Strang splitting and alternating direction implicit methods [175,181].…”

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“…The implication "stability =⇒ convergence" played a central role in the early years of development of numerical methods for PDEs [175,Sec. 3.5].…”

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A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

147

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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“…The semi-discrete numerical approximation is based on the Leap-Frog method (e.g., [13][14][15][16]), an explicit second order in time method with uniform time step dt, that reads:…”

Section: The 2d Acoustic Seismic Equation and Its Numerical Modelmentioning

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A local adaptive method for the numerical approximation in seismic wave modelling

Galuzzi

Zampieri

Stucchi

2017

Communications in Applied and Industrial Mathematics

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We propose a new numerical approach for the solution of the 2D acoustic wave equation to model the predicted data in the field of active-source seismic inverse problems. This method consists in using an explicit finite difference technique with an adaptive order of approximation of the spatial derivatives that takes into account the local velocity at the grid nodes. Testing our method to simulate the recorded seismograms in a marine seismic acquisition, we found that the low computational time and the low approximation error of the proposed approach make it suitable in the context of seismic inversion problems.

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Difference Methods for Initial-Value Problems

Cited by 1,762 publications

References 0 publications

On certain finite difference schemes for hyperbolic systems

On certain finite difference schemes for hyperbolic systems

A review of numerical methods for nonlinear partial differential equations

A local adaptive method for the numerical approximation in seismic wave modelling

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