Generalised Riemann invariants

Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

Abstract. The Generalized Riemann Problem (GRP) for nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resoluti...

The generalised Riemann problem: the basis of ADER schemes

Advection-reaction type partial differential equations model a wide variety of phenomena in several disciplines in physics, chemistry, environmental sciences, geometry, financial mathematics and many others. Generally, these non-linear inhomo-geneous equations must be solved in complicated multi-dimensional domains and thus analytical solutions are only available under very special circumstance...

Gromov Invariants for Holomorphic Maps from Riemann Surfaces to Grassmannians

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme. The latter may be embedded into the moduli space of solutions to a generalized ve...

Absolute Scalar Invariants and the Isometric Correspondence of Riemann Spaces.

Geometric Interpretation of the Mixed Invariants of the Riemann Spinor

Mixed invariants are used to classify the Riemann spinor in the case of Einstein-Maxwell fields and perfect fluids. In the EinsteinMaxwell case these mixed invariants provide information as to the relative orientation of the gravitational and electromagnetic principal null directions. Consideration of the perfect fluid case leads to some results about the behaviour of the Bel-Robinson tensor re...