Solving first-order initial-value problems by using an explicit non-standard A-stable one-step method in variable step-size formulation

This paper presents the construction of a new family of explicit schemes for the numerical solution of initial-value problems of ordinary differential equations (ODEs). The one-parameter family is constructed by considering a suitable rational approximation to the theoretical solution, resulting a family with second-order convergence and A-stable. Imposing that the principal term in the local truncation error vanishes, we obtain an expression for the parameter value in terms of the point (xn, yn) on each step. With this approach, the resulting method has third order convergence maintaining the characteristic of A-stability. Finally, combining this last method with other of order two in order to get an estimation for the local truncation error, an implementation in variable step-size has been considered. The proposed method can be used in a wide range of problems, for solving numerically a scalar ODE or a system of first order ODEs. Several numerical examples are given to illustrate the efficiency and performance of the proposed method in comparison with some existing methods in the literature.

[1]  A. Sedgwick,et al.  An effective variable-order variable-step adams method. , 1973 .

[2]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[3]  Higinio Ramos,et al.  A non-standard explicit integration scheme for initial-value problems , 2007, Appl. Math. Comput..

[4]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[6]  M. Calvo,et al.  On the stability of rational Runge Kutta methods , 1982 .

[7]  H. A. Watts.  Starting step size for an ODE solver , 1983 .

[8]  F. V. Niekerk.  Rational one-step methods for initial value problems , 1988 .

[9]  J. Lambert.  Computational Methods in Ordinary Differential Equations , 1973 .

[10]  Kendall E. Atkinson.  An introduction to numerical analysis , 1978 .

[11]  Higinio Ramos,et al.  Contributions to the development of differential systems exactly solved by multistep finite-difference schemes , 2010, Appl. Math. Comput..

[12]  René Alt,et al.  A-stable one-step methods with step-size control for stiff systems of ordinary differential equations , 1978 .

[13]  J. R. E. O’Malley.  Singular perturbation methods for ordinary differential equations , 1991 .

[14]  Arieh Iserles,et al.  Numerical Solution of Differential Equations , 2006 .

[15]  Arieh Iserles,et al.  A First Course in the Numerical Analysis of Differential Equations: The diffusion equation , 2008 .

[16]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[17]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[18]  L. Shampine,et al.  Computer solution of ordinary differential equations : the initial value problem , 1975 .

[19]  W. E. Milne.  Numerical Solution of Differential Equations , 1953 .

[20]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[21]  G. Sottas,et al.  Rational Runge-Kutta methods are not suitable for stiff systems of ODEs , 1984 .

[22]  Ali Hassan Mohd Murid,et al.  Explicit methods in solving stiff ordinary differential equations , 2004, Int. J. Comput. Math..

[23]  E. El-Zahar.  A Non-Linear Absolutely-Stable Explicit Numerical Integration Algorithm for Stiff Initial-Value Problems , 2013 .

[24]  Jianlin Xia,et al.  Two low accuracy methods for stiff systems , 2001, Appl. Math. Comput..

[25]  E. Hairer.  Unconditionally stable explicit methods for parabolic equations , 1980 .

[26]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[27]  R. Jain,et al.  Numerical Methods for Scientific and Engineering Computation , 1985 .

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

[29]  Peter Deuflhard,et al.  Newton Methods for Nonlinear Problems , 2004 .

[30]  H. H. Rosenbrock,et al.  Some general implicit processes for the numerical solution of differential equations , 1963, Comput. J..

[31]  A Nonlinear Explicit One‐Step Integration Scheme for Singular Autonomous Initial Value Problems , 2007 .

[32]  Manuel Calvo,et al.  On the change of step size in multistep codes , 1993, Numerical Algorithms.

[33]  J. D. Lambert,et al.  Nonlinear methods for stiff systems of ordinary differential equations , 1974 .