Eecient Characteristic Projection in Upwind Diierence Schemes for Hyperbolic Systems (the Complementary Projection Method)

The standard construction of upwind diierence schemes for hyper-bolic systems of conservation laws requires the full eigensystem of the Jacobian matrix. This system is used to deene the transformation into and out of the characteristic scalar elds, where upwind diierencing 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 sub-space. 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 diierencing without specifying a basis. For systems with eigenvalues of high multiplicity, this approach simpliies the analytical and programming eeort and reduces the computational cost. Numerical experiments show no signiicant diierence in computed results between this formulation and the traditional one, and thus we recommend its use for these types of problems. 1 This complementary projection method has other applications. For example, it can be used to extend upwind schemes to some weakly hy-perbolic systems. These lack complete eigensystems, so the traditional form of characteristic upwinding is not possible.