An Efficient Finite Difference Scheme for the 2D Sine-Gordon Equation
For researchers solving nonlinear wave equations, this provides an efficient and energy-conserving numerical method, though it is an incremental improvement over existing finite difference approaches.
The paper proposes a second-order finite difference scheme for the 2D sine-Gordon equation that preserves discrete energy conservation for the undamped case. Numerical simulations verify second-order accuracy and robustness.
We present an efficient second-order finite difference scheme for solving the 2D sine-Gordon equation, which can inherit the discrete energy conservation for the undamped model theoretically. Due to the semi-implicit treatment for the nonlinear term, it leads to a sequence of nonlinear coupled equations. We use a linear iteration algorithm, which can solve them efficiently, and the contraction mapping property is also proven. Based on truncation errors of the numerical scheme, the convergence analysis in the discrete $l^2$-norm is investigated in detail. Moreover, we carry out various numerical simulations, such as verifications of the second order accuracy, tests of energy conservation and circular ring solitons, to demonstrate the efficiency and the robustness of the proposed scheme.