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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, and alternative measures of stability are needed, leading to the introduction of concepts such as total variation diminishing (TVD) and local extremum diminishing (LED) schemes.
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