Waveform Krylov Subspace Methods on a Massively Parallel Computer

Recently, the waveform generalized minimal residual method (WGMRES) was proposed for solving differential-algebraic equations problems. Based on this, several waveform Krylov subspace methods are developed for comparison. Particularly, we propose using an adjoint operator for the waveform bi-conjugate gradient method and the waveform quasi-minimal residual method. The difficulties of developing the adjoint operator will be addressed. Furthermore, these methods are applied to solve a large sparse linear system of ordinary differential equations arising from a parabolic partial differential equation on a DECmpp 12000/Sx parallel computer for comparison. Numerical results show that the WGMRES method and the waveform bi-conjugate gradient stabilized method can achieve better performance than the conventional waveform relaxation methods.