A New Hybrid Method for Bilinear Order Reduction of Burgers Equation based on Balanced Truncation and Iterative Rational Krylov Algorithms

This paper proposes a hybrid method for order reduction of bilinear system model of Burgers' equation, via balanced truncation (BT) and bilinear iterative rational Krylov algorithm (BIRKA). The Monte Carlo simulations demonstrates that by choosing the initial guess randomly, the probability of convergence of the bilinear iterative rational Krylov subspace algorithm method to order reduction of the Burgers equation is only 41%. In the proposed method, at first, the order of the reduced model is determined by the concept of Hankel singular values and by minimizing the integral square of the error index. Then, an initial guess of the reduced bilinear system is obtained by two approaches of Bilinear Balanced Truncation (BBT) and Linear Balanced Truncation (LBT) to ensure convergence of the BIRKA. Output of BBT is a good stable initial guess for BIRKA, but imposes computational complexity of solving generalized Lyapunov equations to find its solution. LBT decreases the computational complexity by providing the initial guess via solving Lyapunov equations. To further decrease the complexity, condition number is replaced instead of eigenvalues in BIRKA. Finally, performance of the proposed method is compared with some classical methods.