Linearized supplementary variable method (LSVM) for thermodynamically consistent partial differential equations

Abstract

[1] Thermodynamically consistent partial differential equations (TCPDEs) arise broadly in science and engineering. Preserving their energy dissipation structure at the discrete level is important for robustness, but is computationally challenging due to the stiff nonlinearities in TCPDEs. The difficulty grows when the mobility operator is state-dependent. In this setting, reformulation-based approaches such as IEQ and SAV typically yield variable-coefficient linear systems and lose their main computational advantage. We propose a linearized supplementary variable method (LSVM) for TCPDEs with a broad class of state-dependent mobilities that admit a splitting into a constant-coefficient core and a state-dependent remainder. The method introduces a balancing mechanism that enforces the energy law. The resulting second-order Crank--Nicolson and BDF2 schemes are constant-coefficient and linear, requiring only the solution of a small number of linear problems with a fixed, prefactorable operator, plus an additional 2 \ttimes 2 linear system per time step. We prove energy-law preservation and unique solvability. Numerical experiments on the Gross--Pitaevskii equation, the molecular beam epitaxy model with slope selection, and the Cahn--Hilliard equation with variable mobility show that the proposed schemes permit significantly larger time steps than many existing implicit methods, while maintaining comparable accuracy. We observe substantial efficiency gains in all test problems.