Computing Optimal Forcing Using Laplace Preconditioning (Communications in Computational Physics, 2017)
For problems governed by a non-normal operator, the leading eigenvalueof the operator is of limited interest and a more relevant measure of the stability is obtainedby considering the harmonic forcing causing the largest system response. Variousmethods for determining this so-called optimal forcing exist, but they all sufferfrom great computational expense and are hence not practical for large-scale problems.In the present paper a new method is presented, which is applicable to problemsof arbitrary size. The method does not rely on time-stepping, but on the solution oflinear systems, in which the inverse Laplacian acts as a preconditioner. By formulatingthe search for the optimal forcing as an eigenvalue problem based on the resolventoperator, repeated system solves amount to power iterations, in which the dominanteigenvalue is seen to correspond to the energy amplification in a system for a givenfrequency, and the eigenfunction to the corresponding forcing function. Implementationof the method requires only minor modifications of an existing time-steppingcode, and is applicable to any partial differential equation containing the Laplacian,such as the Navier-Stokes equations. We discuss the method, first, in the context ofthe linear Ginzburg-Landau equation and then, the two-dimensional lid-driven cavityflow governed by the Navier-Stokes equations. Most importantly, we demonstrate thatfor the lid-driven cavity, the optimal forcing can be computed using a factor of up to500 times fewer operator evaluations than the standard method based on exponentialtime-stepping.