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connection between the step-sizes and numerical-speed |
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January 21, 2020, 09:00 |
connection between the step-sizes and numerical-speed
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#1 |
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Prabhat Mishra
Join Date: Jan 2020
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Is there any connection between the step-size and the wave speed ?
for example, we have hyperbolic conservation equation u_t+f(u)_x=0, If we write it's numerical scheme for non-uniform (2p+1) points then how we will write the expression for f'(u) for non-uniform mesh ? I would also like to know of some references regarding this. Last edited by Prabhat Mishra; January 21, 2020 at 12:58. |
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January 21, 2020, 11:38 |
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#2 |
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Filippo Maria Denaro
Join Date: Jul 2010
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Not clear to me why do you define "numerical" a term that appears in the PDE ... Could you detail better you question? Is that focused on hyperbolic equations?
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January 21, 2020, 13:23 |
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#3 |
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Prabhat Mishra
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Yes Sir, the question is based on the hyperbolic equation. In the case of uniform mesh the finite difference scheme under consideration are of (2p+1) points in conservative form,
v_j^{n+1}=H(v_{j-p}^n, ... , v_{j+p}^n):=v_j^n-k/h [F(v_{j-p+1}, ... , v_{j+p}^n) - F(v_{j-p}^n, ... ,v_{j+p-1})]. The expression for a(u) is mentioned in Page No. (303) of the given Reference. Reference: A. Harten, J. M. Hyman and P. D. Lax: On Finite Difference Approximation and Entropy Conditions for shocks How we will write the expression for wave- speed a(u) for non-uniform mesh-points ? |
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January 21, 2020, 19:08 |
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#4 | |
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,760
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Quote:
One of the best textbook I can address is Leveque, Finite Volume for Hyperbolic Problems. In a conservative method you do not write the equation in a quasi linear form, that is you evaluate the flux f(u) not the term df/du*du/dx |
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