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Euler–Cromer Algorithm
In mathematics, the semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a modification of the Euler method for solving Hamilton's equations, a system of ordinary differential equations that arises in classical mechanics. It is a symplectic integrator and hence it yields better results than the standard Euler method. Setting The semi-implicit Euler method can be applied to a pair of differential equations of the form :\begin &= f(t,v) \\ &= g(t,x), \end where ''f'' and ''g'' are given functions. Here, ''x'' and ''v'' may be either scalars or vectors. The equations of motion in Hamiltonian mechanics take this form if the Hamiltonian is of the form : H = T(t,v) + V(t,x). \, The differential equations are to be solved with the initial condition : x(t_0) = x_0, \qquad v(t_0) = v_0. The method The semi-implicit Euler method produces an approximate discrete solution by iterating :\begin v_ & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Integration
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who treated it in his book ''Institutionum calculi integralis'' (published 1768–1870). The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size. The Euler method often serves as the basis to construct more complex methods, e.g., predictor–corrector method. Informal geometrical description Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |