Suppressing the onset of instabilities in the time-domain equivalent source method using a multistep approach.

Numerical instability is an important issue that should be addressed in the time-domain equivalent source method (TESM). This study proposes a multistep method to stabilize TESM when using the measured acoustic pressure data to optimize equivalent source strengths. Unlike the conventional single-step method that solves each time step in the time-marching process of TESM, the proposed method performs a one-time solution for several time steps. The multistep solution can potentially reduce the accumulation rate of error, and improves filtering effects by changing the structure of the matrix that needs to be inverted when the truncated singular value decomposition or Tikhonov regularization is used in the time-marching process. Numerical simulations with three examples demonstrate the effectiveness of the multistep method in improving the stability of solutions compared with the single-step method. Effects of the number of merged time steps on the solutions are also discussed to guide the selection process. Finally, the sensitivity of the multistep method to numerical parameters is investigated to demonstrate its consistency under different configurations of numerical parameters.