Efficient Characteristic Projection in Upwind Difference Schemes for Hyperbolic Systems: The Complementary Projection Method
✍ Scribed by Ronald P. Fedkiw; Barry Merriman; Stanley Osher
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 208 KB
- Volume
- 141
- Category
- Article
- ISSN
- 0021-9991
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✦ Synopsis
The standard construction of upwind difference schemes for hyperbolic systems of conservation laws requires the full eigensystem of the Jacobian matrix. This system is used to define the transformation into and out of the characteristic scalar fields, where upwind differencing is meaningful. When the Jacobian has a repeated eigenvalue, the associated normalized eigenvectors are not uniquely determined, and an arbitrary choice of eigenvectors must be made to span the characteristic subspace. In this report we point out that it is possible to avoid this arbitrary choice entirely. Instead, a complementary projection technique can be used to formulate upwind differencing without specifying a basis. For systems with eigenvalues of high multiplicity, this approach simplifies the analytical and programming effort and reduces the computational cost. Numerical experiments show no significant difference in computed results between this formulation and the traditional one, and thus we recommend its use for these types of problems. This complementary projection method has other applications. For example, it can be used to extend upwind schemes to some weakly hyperbolic systems. These lack complete eigensystems, so the traditional form of characteristic upwinding is not possible.