A comparison of implicit time integration methods for nonlinear relaxation and diffusion

Several time integration methods for nonlinear systems are compared. All of the time discretizations are based on the θ-method, but differ in their treatment of the implicit nonlinear terms. One method converges the implicit nonlinear terms to a small tolerance and is often referred to as nonlinearly consistent (NC). Newtonʹs method, or its approximation Newton–Krylov, is used to converge the nonlinearities. The other methods considered are linearized and comparisons are made for a relaxation problem and a radiation diffusion problem. The linearized one-step method that uses the full Jacobian is shown to have similar accuracy as NC methods. The lagged linearization method and an extension that is second-order accurate are also studied. A truncation error analysis complements the numerical results. For the relaxation problem, it is shown that each of the second-order accurate linearized methods may be more accurate than an NC method, depending on the degree of nonlinearity in the problem. For the radiation diffusion problem, in general the NC method is most accurate and allows a larger time step. However, the linearized methods perform surprisingly well.