Stability for Nonlinear Numerical Scheme

February 1, 2001, 13:34

It is usual for the application of Von Neumann or matrix methods in the stability analysis of a numerical scheme for a linear PDE. But Neuman method can only be used for linear problems. For nonlinear problem, for example Du/Dt = -(1/2)D(uu)/Dx - 32 (Du/Dx)^2+D^2u/Dx^2, Neuman method can not be used. How do we do the stability analysis for this kind of nonliear PDE after disretization, or where can I get reference materials on this issue?

Thank you!

February 6, 2001, 10:59

If you write the discretized non-linear equation assuming a first order perturbation

U=U0+deltaU

you can linearize the equation (after the discretization).

WOuld that help you ?

Patrick

February 7, 2001, 12:51

This problem was addressed by Poinsot and Candel for the inviscid Burgers-Equation in Journal of Computational Physics, vol. 62, 282-296, 1986.

February 12, 2001, 13:21

Culbert Laney discussed this in great detail in Chapters 15 and 16 of his book Computational Gasdynamics. In fact the linear stability is still relevant.

February 1, 2001, 13:34	Stability for Nonlinear Numerical Scheme	#1
Guo Guest Posts: n/a	It is usual for the application of Von Neumann or matrix methods in the stability analysis of a numerical scheme for a linear PDE. But Neuman method can only be used for linear problems. For nonlinear problem, for example Du/Dt = -(1/2)D(uu)/Dx - 32 (Du/Dx)^2+D^2u/Dx^2, Neuman method can not be used. How do we do the stability analysis for this kind of nonliear PDE after disretization, or where can I get reference materials on this issue? Thank you!

February 6, 2001, 10:59	Re: Stability for Nonlinear Numerical Scheme	#2
Patrick Godon Guest Posts: n/a	If you write the discretized non-linear equation assuming a first order perturbation U=U0+deltaU you can linearize the equation (after the discretization). WOuld that help you ? Patrick

February 7, 2001, 12:51	Re: Stability for Nonlinear Numerical Scheme	#3
Ralf Massjung Guest Posts: n/a	This problem was addressed by Poinsot and Candel for the inviscid Burgers-Equation in Journal of Computational Physics, vol. 62, 282-296, 1986.

February 12, 2001, 13:21	Re: Stability for Nonlinear Numerical Scheme	#4
clifford bradford Guest Posts: n/a	Culbert Laney discussed this in great detail in Chapters 15 and 16 of his book Computational Gasdynamics. In fact the linear stability is still relevant.

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