Matrix-free W-methods using a multiple Arnoldi iteration Original Research Article

The standard Arnoldi method approximates the solution of one single linear system through projection onto a Krylov space constructed from the right-hand side vector. The different stage equations of a W-method use the same matrix but different right-hand sides, which are, in general, not part of the original Krylov space. For the case of two right-hand sides we discuss two well-known deterministic strategies that extend the Arnoldi process from the first stage and present a new one that performs Krylov steps adaptively to minimize the residual of the next approximation. An implementation of these methods is tested on three different parabolic problems and compared with the code VODPK.