Solution Interpolation Method for Highly Oscillating Hyperbolic Equations

This paper deals with a novel numerical scheme for hyperbolic equations with rapidly changing terms. We are especially interested in the quasilinear equationut+aux=f(x)u+g(x)unand the wave equationutt=f(x)uxxthat have a highly oscillating term likef(x)=sin(x/ε), ε≪1. It also applies to the equations involving rapidly changing or even discontinuous coefficients. The method is based on the solution interpolation and the underlying idea is to establish a numerical scheme by interpolating numerical data with a parameterized solution of the equation. While the constructed numerical schemes retain the same stability condition, they carry both quantitatively and qualitatively better performances than the standard method.

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GEOMETRIC INTEGRATION BY SOLUTION INTERPOLATION

International Journal of Modern Physics C ◽

10.1142/s0129183109013637 ◽

2009 ◽

Vol 20 (02) ◽

pp. 313-322

Author(s):

PILWON KIM

Keyword(s):

Differential Equation ◽

Heat Equation ◽

Exact Solutions ◽

Standard Method ◽

Interpolation Method ◽

Nonlinear Heat Equation ◽

Geometric Integration ◽

Numerical Schemes ◽

New Class ◽

Geometric Integrators

Numerical schemes that are implemented by interpolation of exact solutions to a differential equation naturally preserve geometric properties of the differential equation. The solution interpolation method can be used for development of a new class of geometric integrators, which generally show better performances than standard method both quantitatively and qualitatively. Several examples including a linear convection equation and a nonlinear heat equation are included.

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Some Remarks on the Stability Condition of Numerical Scheme of the KdV-type Equation

Journal of Mathematics Research ◽

10.5539/jmr.v9n4p11 ◽

2017 ◽

Vol 9 (4) ◽

pp. 11 ◽

Cited By ~ 1

Author(s):

Chun-Te Lee ◽

Jeng-Eng Lin ◽

Chun-Che Lee ◽

Mei-Li Liu

Keyword(s):

Stability Condition ◽

Numerical Scheme ◽

Kdv Equation ◽

Type Equation ◽

Stability Criteria ◽

Numerical Schemes ◽

The Kdv Equation ◽

The Stability ◽

Difference Fourier ◽

Central Finite Difference

This paper has employed a comparative study between the numerical scheme and stability condition. Numerical calculations are carried out based on three different numerical schemes, namely the central finite difference, fourier leap-frog, and fourier spectral RK4 schemes. Stability criteria for different numerical schemes are developed for the KdV equation, and numerical examples are put to test to illustrate the accuracy and stability between the solution profile and numerical scheme.

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Numerical Simulation of Feller’s Diffusion Equation

Mathematics ◽

10.3390/math7111067 ◽

2019 ◽

Vol 7 (11) ◽

pp. 1067

Author(s):

Denys Dutykh

Keyword(s):

Numerical Simulation ◽

Diffusion Coefficient ◽

Diffusion Equation ◽

Fluid Mechanics ◽

Open Source ◽

Stability Condition ◽

Numerical Scheme ◽

Turbulence Theory ◽

Wave Turbulence ◽

Matlab Code

This article is devoted to Feller’s diffusion equation, which arises naturally in probability and physics (e.g., wave turbulence theory). If discretized naively, this equation may represent serious numerical difficulties since the diffusion coefficient is practically unbounded and most of its solutions are weakly divergent at the origin. In order to overcome these difficulties, we reformulate this equation using some ideas from the Lagrangian fluid mechanics. This allows us to obtain a numerical scheme with a rather generous stability condition. Finally, the algorithm admits an elegant implementation, and the corresponding Matlab code is provided with this article under an open source license.

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The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics

Abstract and Applied Analysis ◽

10.1155/s1085337596000103 ◽

1996 ◽

Vol 1 (2) ◽

pp. 203-217 ◽

Cited By ~ 34

Author(s):

George Avalos

Keyword(s):

Wave Equation ◽

Feedback Control ◽

Exponential Stability ◽

Parabolic System ◽

Hyperbolic Equations ◽

Coupled System ◽

Beam Equation ◽

Structural Acoustics ◽

Uniform Stabilization ◽

Structure Interaction

We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDE's which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the interior of a bounded domainΩ, coupled to a “parabolic–like” beam equation holding on∂Ω, and wherein the coupling is accomplished through velocity terms on the boundary. Our result is an analog of a recent result by Lasiecka and Triggiani which shows the exponential stability of the wave equation via Neumann feedback control, and like that work, depends upon a trace regularity estimate for solutions of hyperbolic equations.

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