Reduced order modeling for fluid flows subject to quadratic type nonlinearities

Explicit model reduction for nonlinear systems with no prior information about the type of nonlinearity involved is difficult and challenging. It is easier to reduce nonlinear systems which nonlinearity is known. In this paper we introduce two nonlinear model reduction techniques for quadratic nonlinear systems. The first technique is nonlinear balanced truncation. The Controlability and observability gramians are computed by solving the Hamilton Jacobi equations and then used to find the transformation function to get the nonlinear balanced truncated system. The second technique is using Arnoldi algorithm. We apply both techniques to a practical nonlinear quadratic system which is the two-dimensional Burgers equation problem of a fluid passing an obstacle.