Numerical comparison of modified-energy stable SAV-type schemes and classical BDF methods on benchmark problems for the functionalized Cahn-Hilliard equation

Abstract

In this paper, we construct and test a class of linear numerical schemes for the Functionalized Cahn-Hilliard (FCH) equation with a symmetric double-well potential function by using a stabilized scalar auxiliary variable (SAV) method. To get a fair assessment of these new SAV-type schemes, we compare output with numerical solutions obtained by the classical, fully-implicit BDF1 and BDF2 schemes. We prove the unconditional unique solvability of the SAV systems and demonstrate the detailed steps used for finding the solutions to these systems. Two sixth-order constant-coefficient linear equations need to be solved at each time step for every SAV scheme. We also provide a theoretical analysis of the unconditional modified-energy stability for the schemes using the usual tools. The Fourier pseudo-spectral method is used as the spatial discretization. Several numerical tests are performed to verify the theoretical analyses and to compute some interesting problems that are physically relevant. Simulations of phase separation in 2D and 3D show the schemes can capture the correct qualitative dynamical behavior and, at the same time, the original physical FCH free energy is dissipated. The classical BDF1 (backward Euler) and BDF2 fully implicit methods, which have significantly smaller local truncation errors (LTEs), are used to repeat several numerical calculations and give a more objective measure of the accuracy and efficiency of the SAV schemes. To keep things simple and fair, for this preliminary battery of comparison tests, we use only fixed, uniform time step sizes. In this setting, the SAV schemes often have an advantage in terms of computational efficiency, being up to three times faster in CPU time when a relatively large time step size is used. However, when accuracy is counted in the measures of computational efficiency, the classical BDF methods often perform better than the linear SAV methods, with an advantage of up to three digits of precision. If the final target of a computation is a relatively high global accuracy, then the method with the least computational time to achieve that accuracy is very often classical BDF2. But, these conclusions are not universal; some test results are subtle and ambiguous. In any case, while SAV methods can be constructed in such a way that they are both energy stable and accurate, they are, however, not always a good choice in practical, real-world computations, because their large LTEs can severely limit their true efficiency.