This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in MATLAB.

The MATLAB ODE Suite

A family of embedded Runge-Kutta formulae

This comprehensive text and reference work on numerical weather prediction, first published in 2002, covers not only methods for numerical modeling, but also the important related areas of data assimilation and predictability. It incorporates all aspects of environmental computer modeling including an historical overview of the subject, equations of motion and their approximations, a modern and clear description of numerical methods, and the determination of initial conditions using weather observations (an important science known as data assimilation). Finally, this book provides a clear discussion of the problems of predictability and chaos in dynamical systems and how they can be applied to atmospheric and oceanic systems. Professors and students in meteorology, atmospheric science, oceanography, hydrology and environmental science will find much to interest them in this book, which can also form the basis of one or more graduate-level courses.

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Atmospheric Modeling, Data Assimilation and Predictability

Finite difference approximations -- Steady states and boundary value problems -- Elliptic equations -- Iterative methods for sparse linear systems -- The initial value problem for ordinary differential equations -- Zero-stability and convergence for initial value problems -- Absolute stability for ordinary differential equations -- Stiff ordinary differential equations -- Diffusion equations and parabolic problems -- Addiction equations and hyperbolic systems -- Mixed equations -- Appendixes: A. Measuring errors -- B. Polynomial interpolation and orthogonal polynomials -- C. Eigenvalues and inner-product norms -- D. Matrix powers and exponentials -- E. Partial differential equations.

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Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-dependent Problems

In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines approach. The differential equations can be represented in R code or as compiled code. In the latter case, R is used as a tool to trigger the integration and post-process the results, which facilitates model development and application, whilst the compiled code significantly increases simulation speed. The methods implemented are efficient, robust, and well documented public-domain Fortran routines. They include four integrators from the ODEPACK package (LSODE, LSODES, LSODA, LSODAR), DVODE and DASPK2.0. In addition, a suite of Runge-Kutta integrators and special-purpose solvers to efficiently integrate 1-, 2- and 3-dimensional partial differential equations are available. The routines solve both stiff and non-stiff systems, and include many options, e.g., to deal in an efficient way with the sparsity of the Jacobian matrix, or finding the root of equations. In this article, our objectives are threefold: (1) to demonstrate the potential of using R for dynamic modeling, (2) to highlight typical uses of the different methods implemented and (3) to compare the performance of models specified in R code and in compiled code for a number of test cases. These comparisons demonstrate that, if the use of loops is avoided, R code can efficiently integrate problems comprising several thousands of state variables. Nevertheless, the same problem may be solved from 2 to more than 50 times faster by using compiled code compared to an implementation using only R code. Still, amongst the benefits of R are a more flexible and interactive implementation, better readability of the code, and access to R’s high-level procedures. deSolve is the successor of package odesolve which will be deprecated in the future; it is free software and distributed under the GNU General Public License, as part of the R software project.

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Solving Differential Equations in R: Package deSolve

High order embedded Runge-Kutta formulae

(The latter summation is taken as zero when i = 1. Here, xn+1 =xn + hn, where hn = 6(xn)h with 0 < 9(x) « 1; we have taken y0 = y(x0) and y'o =y'(x0), and yn, y'n, yn, and y'n denote approximations to the true solution y(xn) and y'{xn) at the point x =xn. The caps denote the approximations from the gth-order formula. It should be noted that the embedded process is here applied in local-extrapo lation mode or higher-order mode, similar to the situation for RK processes (Dormand

Families of Runge-Kutta-Nystrom Formulae

Description des processus de Runge-Kutta-Nystrom d'ordre eleve pour la solution numerique d'une equation differentielle du second ordre. Resultats de tests

High-Order Embedded Runge-Kutta-Nystrom Formulae

Differential Equations Classification of Differential Equations Linear Equations Non-Linear Equations Existence and Uniqueness of Solutions Numerical Methods Computer Programming First Ideas and Single-Step Methods Analytical and Numerical Solutions A First Example The Taylor Series Method Runge-Kutta Methods Second and Higher Order Equations Error Considerations Definitions Local Truncation Error for the Taylor Series Method Local Truncation Error for the Runge-Kutta Method Local Truncation and Global Errors Local Error and LTE Runge-Kutta Methods Error Criteria A Third Order Formula Fourth Order Formulae Fifth and Higher Order Formulae Rationale for Higher Order Formulae Computational Examples Step-Size Control Steplength Prediction Error Estimation Local Extrapolation Error Estimation with RK Methods More Runge-Kutta Pairs Application of RK Embedding Dense Output Construction of Continuous Extensions Choice of Free Parameters Higher-Order Formulae Computational Aspects of Dense Output Inverse Interpolation Stability and Stiffness Absolute Stability Non-Linear Stability Stiffness Improving the Stability of RK Methods Multistep Methods The Linear Multistep Process Selection of Parameters A Third Order Implicit Formula A Third Order Explicit Formula Predictor-Corrector Schemes Error Estimation A Predictor-Corrector Program Multistep Formulae from Quadrature Quadrature Applied to Differential Equations The Adams-Bashforth Formulae The Adams-Moulton Formulae Other Multistep Formulae Varying the Step Size Numerical Results Stability of Multistep Methods Some Numerical Experiments Zero-Stability Weak Stability Theory Stability Properties of Some Formulae Stability of Predictor-Corrector Pairs Methods for Stiff Systems Differentiation Formulae Implementation of BDF Schemes A BDF Program Implicit Runge-Kutta Methods A Semi-Implicit RK Program Variable Coefficient Multistep Methods Variable Coefficient Integrators Practical Implementation Step-Size Estimation A Modified Approach An Application of STEP90 Global Error Estimation Classical Extrapolation Solving for the Correction An Example of Classical Extrapolation The Correction Technique Global Embedding A Global Embedding Program Second Order Equations Transformation of the RK Process A Direct Approach to the RKNG Processes The Special Second Order Problem Dense Output for RKN Methods Multistep Methods Partial Differential Equations Finite Differences Semi-Discretization of the Heat Equation Highly Stable Explicit Schemes Equations with Two Space Dimensions Non-Linear Equations Hyperbolic Equations Appendix A: Programs for Single Step Methods A Variable Step Taylor Method An Embedded Runge-Kutta Program A Sample RK Data File An Alternative Runge-Kutta Scheme Runge-Kutta with Dense Output A Sample Continuous RK Data File Appendix B: Multistep Programs A Constant Steplength Program A Variable Step Adams PC Scheme A Variable Coefficient Multistep Package Appendix C: Programs for Stiff Systems A BDF Program A Diagonally Implicit RK Program Appendix D: Global Embedding Programs The Gem Global Embedding Code The GEM90 Package with Global Embedding A Driver Program for GEM90 Appendix E: A Runge-Kutta Nystroem Program Bibliography Index Each chapter also includes an introduction and a section of exercise problems.

J. R. Dormand

Papers

A family of embedded Runge-Kutta formulae

High order embedded Runge-Kutta formulae

Families of Runge-Kutta-Nystrom Formulae

High-Order Embedded Runge-Kutta-Nystrom Formulae

Numerical Methods for Differential Equations: A Computational Approach