Journal Article10.1137/0715055
A Variable Mesh Finite Difference Method for Solving a Class of Parabolic Differential Equations in One Space Variable
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TL;DR: In this paper, a variable mesh finite difference scheme for a class of parabolic differential equations which exhibit shock-like structures is developed, and a properly chosen variable mesh will yield results comparable in accuracy to one using a much finer uniform mesh.
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Abstract: A variable mesh finite difference scheme for a class of parabolic differential equations which exhibit shock-like structures is developed. It is shown that a properly chosen variable mesh will yield results comparable in accuracy to one using a much finer uniform mesh. Computable criteria and schemes for generating such variable meshes are given. A scheme is then applied to the Burgers' and modified Burgers’ equations with a small viscosity. Excellent agreement is obtained with known exact solutions.
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Citations
The numerical solution of second-order boundary value problems on nonuniform meshes
TL;DR: It is shown that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order, which illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes.
Stability Restrictions on Second Order, Three Level Finite Difference Schemes for Parabolic Equations
TL;DR: In this paper, the authors examined the stability restrictions for second order schemes using linear stability analysis, and illustrate their behaviour on Burgers' equation, and showed that they are easy to use and apply readily to nonlinear equations.
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On the Numerical Solution of Initial/Boundary-Value Problems in One Space Dimension
TL;DR: In this article, the numerical solution of initial/boundary value problems of the form \[ A(u,x,t)u_t + B(u x,t),u y = c(u y, t), c(x, t, y) is considered, and a mesh selection technique is described that accurately places points in regions where the solution is rapidly changing.
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A local mesh-refinement technique for incompressible flows
TL;DR: A local mesh refinement procedure is introduced as part of a Multi-Grid scheme for the solution of the Navier-Stokes equations in primitive variables, which allows the storage of the dependent variables, without increasing in the required computer memory.
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TL;DR: In this paper, the Navier-Stokes equations for one-dimensional non-stationary flow of a compressible viscous fluid are compared to the shock wave theory of a model of turbulence.
A new method for solving two-point boundary value problems using optimal node distribution
V.E Denny,R.B Landis +1 more
TL;DR: In this paper, a new method for solving two-point boundary value problems by finite difference methods has been developed, based on the observation that local truncation errors associated with central difference analogues of the defining differential equation become arbitrarily small as the interior node points are arranged in an optimal sequence.
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