Energy-stable linear convex splitting methods for the parabolic sine-Gordon equation

We propose a linear convex splitting approach for the parabolic sine-Gordon equation. This linear formulation ensures unique solvability and high computational efficiency. When combined with a convex splitting Runge–Kutta method, it achieves high-order temporal accuracy and unconditional energy stability. For the first-order scheme, we establish the discrete maximum principle, a notable property of the parabolic sine-Gordon equation, although this principle is observed to be numerically violated in the second-order scheme. Spatial discretization is performed employing a standard second-order accurate finite difference method. Numerical experiments are provided to validate the accuracy, energy stability, and dynamic behavior of the proposed schemes.