A method of finding unstable steady solutions by forward time integration: relaxation to a running mean

Proposed is a method of finding unstable steady solutions based on relaxation to a running mean. It works for moderate supercriticality, when the growth rates of the instabilities do not exceed their frequencies. To calculate the running mean one should use the decaying exponential function as a kernel in the convolution formula, it offers a combination of necessary stabilization properties with the efficiency of numerical implementation. The method requires specification of the relaxation TR and averaging TA time scales, such that in the transformed eigenspectrum frequencies above the cutoff 1/TA experience additional damping with 1/TR rate. All originally decaying modes remain decaying. The zero frequency remains unchanged, which implies that the steady states are invariant under the transformation. The acceptable values for TR and TA can be obtained from an estimate of the frequencies and growth rates of the modes leading to instability. For faster convergence to the steady state TR should be set as large as possible, provided it causes sufficient damping of the instability. The numerical implementation of this method requires only very simple modifications to the program used for time integration of the original system.