关键词: 强稳定性, 正性 , 不动点保持, 稳定化技术, 参数方法

Abstract: High-order accurate and stable explicit methods are powerful in solving differential equations efficiently. In this work, we propose a systematic framework to trade off accuracy for stability, especially the unconditional preservation of strong stability, positivity, range boundedness and contractivity. The whole algorithm consists of three steps: (1) Introducing a stabilizing term in the continuous system; (2) Integrating the system using an explicit exponential method; (3) Substituting the exponential functions with suitable approximations. We first show that a class of first- and second-order exponential time difference Runge-Kutta schemes are capable to preserve structures unconditionally when suitable stabilization parameter is chosen. Then by adopting the integrating factor approach with high-order Runge-Kutta and multi-step schemes as underlying schemes, three different approximation techniques are developed to make high-order schemes unconditionally structure-preserving, i.e., (1) a Taylor polynomial approximation; (2) a recursive approximation; (3) an approximation using combinations of exponential and linear functions. The proposed parametric schemes can be deployed to stiff problems straightforwardly by treating the stiff linear term as an integrating factor. The resulting time integration methods retain the explicitness and convergence orders of underlying time-marching schemes, yet with unconditional preservation of structures. The proposed framework using the second and third approximations has relatively mild requirement on underlying schemes, i.e., all coefficients are non-negative. Thus the parametric Runge-Kutta schemes can reach up to the fourth-order, and there is no order barrier in parametric multi-step schemes. The only free parameter--the stablization parameter in the framework can be determined a priori based on the forward Euler conditions. Unlike implicit methods, the parametric methodology allows for solving nonlinear problems stably and explicitly. As an alternative to conditionally structure-preserving methods, the proposed schemes are promising for the efficient computation of stiff and nonlinear problems. Numerical tests on benchmark problems with different stiffness are carried out to assess the performance of parametric methods.

Key words: Strong stability, Positivity , Fixed-point-preserving , Stabilization technique , Parametric method