Numerical exterior algebra and the compound matrix method

Abstract

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 C n , by using compound matrices as coordinates, is reformulated in a coordinate-free way using exterior algebra spaces, k (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 k (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 (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 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

Linear stability analysis of a horizontal phase boundary separating two miscible liquids

“…Its numerical solution is not straightforward due to stiffness caused by extreme (too large or too small) values of the governing nondimensional parameters (large Reynolds and Peclet numbers, and small capillary number). We used two numerical methods to solve the problem, the shooting method supplemented with the Gram-Schmidt orthogonalization [48][49][50][51] and the compound matrix method [52][53][54]. The fourth-order adaptive Runge-Kutta technique was used for numerical integration and the Muller's method was employed for root finding.…”

Front propagation in a phase field model with phase-dependent heat absorption

“…This eigenvalue [2]. Similarly there exists an eigenvalue σ − (µ), which corresponds to an exponentially decaying behaviour at z → −∞, and which is the sum of the two eigenvalues of A(µ) with non-negative real part.…”

“…To be more specific, we use the compound matrices to describe the dynamics of the subspaces U − and U + , as well as the Hodge star operator to match these solutions at z = 0. Also, since our problem possesses a 2-2 splitting of eigenvalues at infinity, we have used the implicit midpoint rule to perform the time integration as for such splitting it preserves the Grassmanian manifold to machine accuracy [2].…”

“…, where η − (µ) is the eigenvector of the matrix −A (2) T associated with the eigenvalue −σ + , see [2,7].…”

“…When using the compound matrix method, the solutions do not lie in a linear space, but rather on a Grassmanian manifold, and the numerical integration scheme should be chosen is such a way that it preserves this manifold. Allen and Bridges [2] have shown that the class of Gauss-Legendre Runge-Kutta (GL-RK) methods preserve the Grassmanian G 2 (C 4 ) to machine accuracy. This result motivates the particular choice of the numerical integrator in this paper.…”