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Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution.
The step size is =. The same illustration for = The midpoint method converges faster than the Euler method, as .. Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs).
"Approximate Solution of Ordinary Differential Equations and Their Systems Through Discrete and Continuous Embedded Runge-Kutta Formulae and Upgrading Their Order". Computers & Mathematics with Applications .
The simplest and oldest one-step method, the explicit Euler method, was published by Leonhard Euler in 1768. After a group of multi-step methods was presented in 1883, Carl Runge, Karl Heun and Wilhelm Kutta developed significant improvements to Euler's method around 1900. These gave rise to the large group of Runge-Kutta methods, which form ...
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The consequence of this difference is that at every step, a system of algebraic equations has to be solved. This increases the computational cost considerably. If a method with s stages is used to solve a differential equation with m components, then the system of algebraic equations has ms components.