Linearly Stabilized Schemes for the Time Integration of Stiff Nonlinear PDEs

Abstract

In many applications, the governing PDE to be solved numerically will contain a stiff component. When this component is linear, an implicit time stepping method that is unencumbered by stability restrictions is preferred. On the other hand, if the stiff component is nonlinear, the complexity and cost per step of using an implicit method is heightened, and explicit methods may be preferred for their simplicity and ease of implementation. In this thesis, we analyze new and existing linearly stabilized schemes for the purpose of integrating stiff nonlinear PDEs in time. These schemes compute the nonlinear term explicitly and, at the cost of solving a linear system with a matrix that is fixed throughout, are unconditionally stable, thus combining the advantages of explicit and implicit methods. Applications are presented to illustrate the use of these methods.

Unconditionally Maximum-Principle-Preserving Parametric Integrating Factor Two-Step Runge-Kutta Schemes for Parabolic Sine-Gordon Equations

“…Because of the particularity of the selected problems, the introduced stabilization term was of the form κ(u−u). Thus, it is meaningful to investigate the p(IF)TSRK1/2 approaches for the mean curvature problem [12] or the Cahn-Hilliard equation [34] where a Laplacian-type stabilization κ∆(u−u) is usually introduced to allow large time-step sizes. Moreover, the parametric schemes could preserve maximum principles of other problems, for example, the classical AC-type equations with either Ginzburg-Landau or Flory-Huggins potentials [19].…”

“…To construct efficient time integrators for general stiff, nonlinear differential equations, three key design principles [2,6,12,17] have been considered previously:…”

“…Next, we tested the spatial accuracy of the FD and FP for the RSLM formulation by choosing the moderate pIFTSRK1(4,5) as the temporal integrator, fixing the time step τ= 2 −10 , and refining the spatial grid from N =2 6 to N =2 9 uniformly. The reference solution was computed with N re f = 2 12 and τ = 2 −10 . As expected, the second-order convergence of the finite-difference discretization, and the spectral accuracy of the Fourier pseudospectral discretization were obtained, as shown in Table 1.…”