Reinterpretation of Multi-Stage Methods for Stiff Systems: A Comprehensive Review on Current Perspectives and Recommendations

In this paper, we compare a multi-step method and a multi-stage method for stiff initial value problems. Traditionally, the multi-step method has been preferred than the multi-stage for a stiff problem, to avoid an enormous amount of computational costs required to solve a massive linear system provided by the linearization of a highly stiff system. We investigate the possibility of usage of multi-stage methods for stiff systems by discussing the difference between the two methods in several numerical experiments. Moreover, the advantages of multi-stage methods are heuristically presented even for nonlinear stiff systems through several numerical tests.

[1]  Dumitru Baleanu,et al.  Effect of microtemperatures for micropolar thermoelastic bodies , 2017 .

[2]  K. Burrage,et al.  Stochastic linear multistep methods for the simulation of chemical kinetics. , 2015, Journal of Chemical Physics.

[3]  Philsu Kim,et al.  A new approach to estimating a numerical solution in the error embedded correction framework , 2018 .

[4]  S. González-Pinto,et al.  Speeding up Netwton-type iterations for stiff problems , 2005 .

[5]  Xiangfan Piao,et al.  One-step L(α)-stable temporal integration for the backward semi-Lagrangian scheme and its application in guiding center problems , 2018, J. Comput. Phys..

[6]  John C. Butcher,et al.  An Iteration Scheme for Implicit Runge—Kutta Methods , 1983 .

[7]  Y. Han,et al.  Solving Implicit Equations Arising from Adams-Moulton Methods , 2002 .

[8]  Fudziah Ismail,et al.  Explicit Integrator of Runge-Kutta Type for Direct Solution of u(4) = f(x, u, u′, u″) , 2019, Symmetry.

[9]  Urs Kirchgraber,et al.  Multi-step methods are essentially one-step methods , 1986 .

[10]  George Em Karniadakis,et al.  Efficient Multistep Methods for Tempered Fractional Calculus: Algorithms and Simulations , 2018, SIAM J. Sci. Comput..

[11]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[12]  Xiangfan Piao,et al.  A Chebyshev Collocation Method for Stiff Initial Value Problems and Its Stability , 2011 .

[13]  T. E. Simos,et al.  New multiple stages multistep method with best possible phase properties for second order initial/boundary value problems , 2018, Journal of Mathematical Chemistry.

[14]  Ralph A. Willoughby,et al.  EFFICIENT INTEGRATION METHODS FOR STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS , 1970 .

[15]  John C. Butcher,et al.  On the implementation of implicit Runge-Kutta methods , 1976 .

[16]  Junghan Kim,et al.  An error embedded method based on generalized Chebyshev polynomials , 2016, J. Comput. Phys..

[17]  C F Curtiss,et al.  Integration of Stiff Equations. , 1952, Proceedings of the National Academy of Sciences of the United States of America.

[18]  Christoph W. Ueberhuber,et al.  Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations , 1977 .

[19]  Soyoon Bak,et al.  High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation , 2019, Numerical Methods for Partial Differential Equations.

[20]  Jingfang Huang,et al.  Accelerating the convergence of spectral deferred correction methods , 2006, J. Comput. Phys..

[21]  A. Prothero,et al.  On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations , 1974 .

[22]  Donato Trigiante,et al.  Energy- and Quadratic Invariants-Preserving Integrators Based upon Gauss Collocation Formulae , 2012, SIAM J. Numer. Anal..

[23]  Xiangfan Piao,et al.  An embedded formula of the Chebyshev collocation method for stiff problems , 2017, J. Comput. Phys..

[24]  Analysis of a class of multi-stage, multistep Runge-Kutta methods , 1994 .

[25]  Theodore A. Bickart,et al.  An Efficient Solution Process for Implicit Runge–Kutta Methods , 1977 .

[26]  G. J. Cooper,et al.  Some schemes for the implementation of implicit Runge-Kutta methods , 1993 .

[27]  Per-Olof Persson,et al.  Stage-parallel fully implicit Runge-Kutta solvers for discontinuous Galerkin fluid simulations , 2017, J. Comput. Phys..

[28]  T. E. Hull,et al.  Comparing numerical methods for stiff systems of O.D.E:s , 1975 .

[29]  G. Dahlquist.  A special stability problem for linear multistep methods , 1963 .

[30]  D. B. Berg,et al.  Three stages symmetric six-step method with eliminated phase-lag and its derivatives for the solution of the Schrödinger equation , 2017, Journal of Mathematical Chemistry.

[31]  G. Dahlquist.  Convergence and stability in the numerical integration of ordinary differential equations , 1956 .