Implicit Euler method

The implicit Euler method (according to Leonhard Euler ) (also backwards Euler method ) is a numerical method for solving initial value problems . It is an implicit process, which means that in each step an equation - generally non-linear - has to be solved.

The procedure

To the numerical solution of the initial value problem:

{\ displaystyle {\ dot {x}} = f (t, x), \ quad \ quad x (t_ {0}) = x_ {0}}

for an ordinary differential equation choose a discretization step size , consider the discrete times ${\ displaystyle h> 0}$

{\ displaystyle t_ {k} = t_ {0} + kh, \ quad \ quad k = 0,1,2, \ dots}

and compute the iterated values

{\ displaystyle x_ {k + 1} = x_ {k} + hf (t_ {k + 1}, x_ {k + 1}) \ quad, \ quad k = 0.1, \ dots}

The value is not given here explicitly, but only implicitly , because it appears on both sides of the equation. To calculate , the equation has to be solved in each iteration step, e.g. B. numerically with the Newton method . This problem does not arise with linear systems, since after can be resolved. ${\ displaystyle f (t_ {k + 1}, x_ {k + 1})}$ ${\ displaystyle x_ {k + 1}}$ ${\ displaystyle x_ {k + 1}}$ ${\ displaystyle x_ {k + 1}}$

The values then represent approximations to the actual values of the exact solution of the initial value problem. The smaller the step size selected, the more computing work has to be done, but the better the approximated values also become. ${\ displaystyle x_ {k}}$ ${\ displaystyle x (t_ {k})}$ ${\ displaystyle h}$

If a method is defined via , one obtains the explicit Euler method . ${\ displaystyle x_ {k + 1} = x_ {k} + hf (t_ {k}, x_ {k})}$

properties

Stability domain of the implicit Euler method.

The implicit Euler method has consistency and convergence order 1. It is L-stable , so its stability region contains the complete left half-plane of the complex number plane . There are therefore no restrictions on the time steps for the implicit Euler method due to stability restrictions, which offsets the need to solve systems of equations in each step. Because of the low order, it is particularly interesting for problems in which the iteration runs into a stable final state and the accuracy of the intermediate results is not of interest.

literature

E. Hairer, SP Norsett, G. Wanner: Solving Ordinary Differential Equations I , Springer Verlag
M. Hermann: Numerics of ordinary differential equations, initial and boundary value problems , Oldenbourg Verlag, Munich and Vienna, 2004, ISBN 3-486-27606-9

Individual evidence

↑ Martin Hermann: Initial Value Problems and Linear Boundary Value Problems . 2nd Edition. DE GRUYTER, 2017, ISBN 978-3-11-050036-3 , p. 16-17 ( limited preview in Google Book search).

[1] Martin Hermann: Initial Value Problems and Linear Boundary Value Problems . 2nd Edition. DE GRUYTER, 2017, ISBN 978-3-11-050036-3 , p. 16-17 ( limited preview in Google Book search).