TY - JOUR
T1 - The apparent numerical stability of unphysical shocks
AU - Sever, Michael
PY - 1986/4/1
Y1 - 1986/4/1
N2 - Difference schemes for hyperbolic systems of conservation laws occasionally converge to an unphysical weak solution, i.e. a weak solution containing discontinuities for which the entropy condition is violated. These unphysical discontinuities, when they exist as solutions of the difference scheme, tend to exhibit a surprising stability under perturbations. We point out here that this can be explained by an energy inequality, which is valid for the discrete approximation but which is not valid as applied to the differential equation itself. In spite of this difficulty, many simple difference schemes are highly successful at converging to the physically correct weak solution. A mechanism for this is given; we show that for shocks of moderate strength, there are numerous quadratic forms in the dependent variables which can serve effectively as entropy functions, i.e. for which an inequality exists determining the physically correct weak solution. It is shown how the limits of the solutions of a difference scheme will often necessarily satisfy such an inequality; as they are generally weak solutions of the given system, they must thus be the correct weak solutions.
AB - Difference schemes for hyperbolic systems of conservation laws occasionally converge to an unphysical weak solution, i.e. a weak solution containing discontinuities for which the entropy condition is violated. These unphysical discontinuities, when they exist as solutions of the difference scheme, tend to exhibit a surprising stability under perturbations. We point out here that this can be explained by an energy inequality, which is valid for the discrete approximation but which is not valid as applied to the differential equation itself. In spite of this difficulty, many simple difference schemes are highly successful at converging to the physically correct weak solution. A mechanism for this is given; we show that for shocks of moderate strength, there are numerous quadratic forms in the dependent variables which can serve effectively as entropy functions, i.e. for which an inequality exists determining the physically correct weak solution. It is shown how the limits of the solutions of a difference scheme will often necessarily satisfy such an inequality; as they are generally weak solutions of the given system, they must thus be the correct weak solutions.
UR - http://www.scopus.com/inward/record.url?scp=84949394573&partnerID=8YFLogxK
U2 - 10.1108/eb010027
DO - 10.1108/eb010027
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AN - SCOPUS:84949394573
SN - 0332-1649
VL - 5
SP - 207
EP - 213
JO - COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering
JF - COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering
IS - 4
ER -