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Numerical Solution of Initial Value Problems

Davoud Mirzaei

TL;DR

This work surveys the numerical solution of initial-value problems for ordinary differential equations, tracing from basic IVP formulation and stability concepts to a broad spectrum of numerical methods. It juxtaposes explicit and implicit one-step methods, Taylor-series and Runge–Kutta schemes, with a rigorous treatment of error, zero-stability, and absolute stability, before addressing stiffness and the deployment of A- and L-stable strategies. The text then extends to multistep methods (Adams–Bashforth/Adams–Moulton/BDF), their convergence and stability properties, and practical considerations such as adaptive time stepping and embedded methods. Practical guidance is provided through discussions of MATLAB's ODE suite and concrete stiffness examples, illustrating the trade-offs between accuracy, stability, and computational cost in solving IVPs. The key message is that the choice of method hinges on problem stiffness, desired accuracy, and computational resources, with implicit and multistep approaches offering robust options for challenging, stiff dynamics.

Abstract

Welcome to a beautiful subject in scientific computing: numerical solution of ordinary differential equations (ODEs) with initial conditions.

Numerical Solution of Initial Value Problems

TL;DR

This work surveys the numerical solution of initial-value problems for ordinary differential equations, tracing from basic IVP formulation and stability concepts to a broad spectrum of numerical methods. It juxtaposes explicit and implicit one-step methods, Taylor-series and Runge–Kutta schemes, with a rigorous treatment of error, zero-stability, and absolute stability, before addressing stiffness and the deployment of A- and L-stable strategies. The text then extends to multistep methods (Adams–Bashforth/Adams–Moulton/BDF), their convergence and stability properties, and practical considerations such as adaptive time stepping and embedded methods. Practical guidance is provided through discussions of MATLAB's ODE suite and concrete stiffness examples, illustrating the trade-offs between accuracy, stability, and computational cost in solving IVPs. The key message is that the choice of method hinges on problem stiffness, desired accuracy, and computational resources, with implicit and multistep approaches offering robust options for challenging, stiff dynamics.

Abstract

Welcome to a beautiful subject in scientific computing: numerical solution of ordinary differential equations (ODEs) with initial conditions.
Paper Structure (32 sections, 4 theorems, 308 equations, 23 figures, 6 tables)

This paper contains 32 sections, 4 theorems, 308 equations, 23 figures, 6 tables.

Table of Contents

  1. An introduction to ODEs
  2. Modelling with ODEs: a funny example
  3. Higher order ODEs
  4. First order system of ODEs
  5. Stability of solutions
  6. Basic numerical methods
  7. Euler's method
  8. Error analysis of Euler's method
  9. General explicit one-step methods
  10. Zero-stability
  11. Absolute stability
  12. Implicit methods
  13. Taylor series methods
  14. Runge-Kutta methods
  15. Stiff differential equations
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $D$ be an open and connected set in $\mathbb{R}^2$ and $f(t,y)$ be a continuous function in both $t$ and $y$ in $D$, and let $(t_0,y_0)$ be an interior point in $D$. Assume that $f$ satisfies the Lipschitz continuity in its second argument, i.e., there exists a constant $L\geqslant 0$ such that Then there exists a unique function $y(t)$ defined on an interval $[t_0-\beta,t_0+\beta]$ for some

Figures (23)

  • Figure 1: Integration curves
  • Figure 2: Three sample solutions for IVP $y'=2\sqrt y$ with $y(0)=0$ (non-uniqueness).
  • Figure 3: Dog chasing a jogger. (left image from freepik.com)
  • Figure 4: Jogging path and the numerically computed trajectory of the dog: The jogger running in a straight path (top-left), the jogger notices the dog and tries to run back (top-right), the jogger running on a circular track (bottom-left), the jogger running on a circular track but the dog is slow (bottom-right).
  • Figure 5: Solutions of a chemical reaction kinetics problem.
  • ...and 18 more figures

Theorems & Definitions (35)

  • Example 1.1
  • Example 1.2
  • Theorem 1.1
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6: Chemical Reaction Kinetics
  • Example 1.7
  • Example 1.8
  • Example 1.9
  • ...and 25 more