MQ拟插值算子求解Sine-Gordon方程的拉格朗日乘子方法

Lagrange Multiplier Method Based on MQ Quasi-Interpolation Operator for Solving Sine-Gordon Equation

摘要: 为提高算法的长时间仿真能力，基于三次MQ拟插值算子构造求解Sine-Gordon方程的一种拉格朗日乘子方法。首先使用拉格朗日乘子法处理边界条件，然后用三次MQ拟插值算子及其导数逼近函数本身及其空间导数，最后用四阶龙格–库塔法进行时间离散。数值实验结果表明拉格朗日乘子方法更能还原问题的非线性特征、稳定性更高和更适合长时间仿真。

Abstract: To improve the long-term simulation capability of existing numerical algorithms, the method of Lagrange multipliers based on cubic MQ quasi-interpolation operator is proposed for solving the Sine-Gordon equation. In this approach, the boundary conditions are first handled using the method of Lagrange multipliers. Subsequently, the cubic MQ quasi-interpolation operator and its derivatives are employed to approximate the solution and its spatial derivatives. Temporal discretization is then performed using the fourth-order Runge-Kutta scheme. The numerical results demonstrate that the method of Lagrange multipliers more effectively captures the nonlinear characteristics of the equation, achieves higher stability, and is particularly well suited for long-term simulations.