Papers โบ Semi-implicit-explicit Runge-Kutta method for nonlinear differential equations
Semi-implicit-explicit Runge-Kutta method for nonlinear differential equations
Lingyun Ding
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A semi-implicit-explicit (semi-IMEX) Runge-Kutta (RK) method is proposed for the numerical integration of ordinary differential equations (ODEs) of the form ๐ฎโฒ = ๐(t,๐ฎ) + G(t,๐ฎ) ๐ฎ, where ๐ is a non-stiff term and G๐ฎ represents the stiff terms. Such systems frequently arise from spatial discretizations of time-dependent nonlinear partial differential equations (PDEs). For instance, G could involve higher-order derivative terms with nonlinear coefficients. Traditional IMEX-RK methods, which treat ๐ explicitly and G๐ฎ implicitly, require solving nonlinear systems at each time step when G depends on ๐ฎ, leading to increased computational cost and complexity. In contrast, the proposed semi-IMEX scheme treats G explicitly while keeping ๐ฎ implicit, reducing the problem to solving only linear systems. This approach eliminates the need to compute Jacobians while preserving the stability advantages of implicit methods. A family of semi-IMEX RK schemes with varying orders of accuracy is introduced. Numerical simulations for various nonlinear equations, including nonlinear diffusion models, the Navier-Stokes equations, and the Cahn-Hilliard equation, confirm the expected convergence rates and demonstrate that the proposed method allows for larger time step sizes without triggering stability issues.
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