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Upwinding in the method of lines. (English) Zbl 0977.68022
Summary: The Method Of Line (MOL) is a procedure for the numerical integration of Partial Differential Equations (PDEs). Briefly, the spatial (boundary value) derivatives of the PDEs are approximated algebraically using, for example, finite differences. If the PDEs have only one initial value variable, typically time, then a system of initial value ordinary differential equations results through the algebraic approximation of the spatial derivatives.
If the PDEs are strongly convective (strongly hyperbolic), they can propagate sharp fronts and even discontinuities, which are difficult to resolve in space. Experience has demonstrated that for these systems, some form of upwinding is generally required when replacing the spatial derivatives with algebraic approximations. Here, we investigate the performance of various forms of upwinding to provide some guidance in the selection of upwind methods in the MOL solution of strongly convective PDEs.

MSC:
68N99 Theory of software
65N40 Method of lines for boundary value problems involving PDEs
Keywords:
method of lines
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References:
[1] Sethian, J.A., Fast marching methods, SIAM rev., 41, 2, 199-235, (1999) · Zbl 0926.65106
[2] W.E. Schiesser, The Numerical Methods of Lines Integration of Partial Differential Equations, Academic Press, San Diego, 1991. · Zbl 0763.65076
[3] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, Berlin, 1997, pp. 426-434.
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