Numerical exterior algebra and the compound matrix method

Summary.

The compound matrix method, which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k=2,3 dimensional subspaces of ${\mathbb C}^n$, by using compound matrices as coordinates, is reformulated in a coordinate-free way using exterior algebra spaces, $\bigwedge^{k}({\mathbb C}^n)$.

This formulation leads to a general framework for studying systems of differential equations on k-dimensional subspaces. The framework requires the development of several new ideas: the role of Hodge duality and the Hodge star operator in the construction, an efficient strategy for constructing the induced differential equations on $\bigwedge^{k}({\mathbb C}^n)$, general formulation of induced boundary conditions, the role of geometric integrators for preserving the manifold of k-dimensional subspaces – the Grassmann manifold, $G_k({\mathbb C}^n)$, and a formulation for induced systems on an unbounded interval.

The numerical exterior algebra framework is most advantageous for numerical solution of differential eigenvalue problems on unbounded domains, where there are significant difficulties in setting up matrix discretizations.

The formulation is presented for k-dimensional subspaces of systems on ${\mathbb C}^n$ with k and n arbitrary, and examples are given for the cases of k=2 and n=4, and k=3 and n=6, with an indication of implementation details for systems of larger dimension.

The theory is illustrated by application to four differential eigenvalue problems on unbounded intervals: hydrodynamic stablity of boundary-layer flow past a compliant surface, the eigenvalue problem associated with the stability of solitary waves, the stability of Bickley jet in oceanography, and the eigenvalue problem associated with the stability of the Ekman layer in atmospheric dynamics.