An input-independent method for solving weakly nonlinear partial differential equations in the frequency domain: application to the Euler equations

Abstract

In this work a novel approach in determining the first and second order frequencydomain Volterra kernels for weakly nonlinear partial differential equations (PDEs) in semidiscrete form based on the application of the harmonic probing (HP) method is presented. This represents a formal extension of the linearized-frequency domain (LFD) methods to a nonlinear framework, leading to a so-called LFD2 method. The method allows for the representation of weak nonlinearities by solving two input-independent linear algebraic systems of equations in the frequency domain and thus circumvents the solution of the nonlinear PDE by numerical integration for each different input, representing a nonlinear reduced-order model (ROM) for the physical phenomena. The general form of the equations is derived and an application to the well known viscous Burgers' equation to show its suitability in representing the nonlinear convective term is shown. Next, an application to the compressible quasi one-dimensional unsteady flow described by the Euler equations is presented. The proposed method overcomes two constraints present in other methods for the solution of nonlinear PDEs, namely, the consideration of exclusively periodic solutions as in the harmonic balance (HB) method and the dependency of the kernels with the input signal as in the Volterra kernel identification methods.