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4 - Fundamental Stability Theory
Antony Jameson, Texas A & M University
Book:

Computational Aerodynamics

Published online:

12 August 2022

Print publication:

01 September 2022, pp 97-130
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Summary

Next, the general definitions of consistency, convergence, and stability are introduced in terms of an arbitrary norm, leading to the Lax equivalence theorem that consistent and stable schemes must converge to the true solution in the limit as the mesh interval and time step are reduced toward zero. Then stability in the Euclidean norm is examined, and the von Neumann stability test is introduced as a convenient way to deduce the stability of any linear scheme.
Nonlinear conservation laws generally admit solutions containing discontinuities, such as shock waves in a fluid flow. This motivates the need for difference schemes in conservation form, andalternative measures of stability are needed, leading to the introduction of concepts such as total variation diminishing (TVD) and local extremum diminishing (LED) schemes.

Numerical Solution to the Multi-Term Time Fractional Diffusion Equation in a Finite Domain
Part of
- Partial differential equations, initial value and time-dependent initial-boundary value problems
- Miscellaneous topics - Partial differential equations
Gongsheng Li, Chunlong Sun, Xianzheng Jia, Dianhu Du
Journal:

Numerical Mathematics: Theory, Methods and Applications / Volume 9 / Issue 3 / August 2016

Published online by Cambridge University Press:

20 July 2016, pp. 337-357

Print publication:

August 2016
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This paper deals with numerical solution to the multi-term time fractional diffusion equation in a finite domain. An implicit finite difference scheme is established based on Caputo's definition to the fractional derivatives, and the upper and lower bounds to the spectral radius of the coefficient matrix of the difference scheme are estimated, with which the unconditional stability and convergence are proved. The numerical results demonstrate the effectiveness of the theoretical analysis, and the method and technique can also be applied to other kinds of time/space fractional diffusion equations.

H1 Stability and Convergence of the FE, FV and FD Methods for an Elliptic Equation
Yinnian He, Xinlong Feng
Journal:

East Asian Journal on Applied Mathematics / Volume 3 / Issue 2 / May 2013

Published online by Cambridge University Press:

28 May 2015, pp. 154-170

Print publication:

May 2013
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We obtain the coefficient matrices of the finite element (FE), finite volume (FV) and finite difference (FD) methods based on P1-conforming elements on a quasi-uniform mesh, in order to approximately solve a boundary value problem involving the elliptic Poisson equation. The three methods are shown to possess the same H1-stability and convergence. Some numerical tests are made, to compare the numerical results from the three methods and to review our theoretical results.