On the Structure of Jacobians for Spectral Methods for Nonlinear Partial Differential Equations

When solving nonlinear partial differential equations with periodic boundary conditions, one frequently uses spectral methods. Implicit time-stepping, the computation of steady-state solutions, or the approximation of inertial manifolds lead to nonlinear equations which can be solved by Newton’s method. In this note the structure of the Jacobian matrix is described for typical nonlinearities, namely, those involving products of the unknown function and its derivatives.