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Generalized Finite-Difference Schemes

Blair Swartz and Burton Wendroff
Mathematics of Computation
Vol. 23, No. 105 (Jan., 1969), pp. 37-49
DOI: 10.2307/2005052
Stable URL: http://www.jstor.org/stable/2005052
Page Count: 13
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Generalized Finite-Difference Schemes
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Abstract

Finite-difference schemes for initial boundary-value problems for partial differential equations lead to systems of equations which must be solved at each time step. Other methods also lead to systems of equations. We call a method a generalized finite-difference scheme if the matrix of coefficients of the system is sparse. Galerkin's method, using a local basis, provides unconditionally stable, implicit generalized finite-difference schemes for a large class of linear and nonlinear problems. The equations can be generated by computer program. The schemes will, in general, be not more efficient than standard finite-difference schemes when such standard stable schemes exist. We exhibit a generalized finite-difference scheme for Burgers' equation and solve it with a step function for initial data.

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