Study on Banded Implicit Runge-Kutta Methods for Solving Stiff Differential Equations

The implicit Runge-Kutta method with A-stability is suitable for solving stiff differential equations. However, the fully implicit Runge-Kutta method is very expensive in solving large system problems. Although some implicit Runge-Kutta methods can reduce the cost of computation, their accuracy and stability are also adversely affected. Therefore, an effective banded implicit Runge-Kutta method with high accuracy and high stability is proposed, which reduces the computation cost by changing the Jacobian matrix from a full coefficient matrix to a banded matrix. Numerical solutions and results of stiff equations obtained by the methods involved are compared, and the results show that the banded implicit Runge-Kutta method is advantageous to solve large stiff problems and conducive to the development of simulation.