It can be assigned to both the Runge-Kutta method and the Adams-Moulton method . The trapezoidal method is A-stable with the special feature that no amplitude error occurs for the oscillation equation. The procedure can be derived from the trapezoidal rule :
Now the idea is to use a simple quadrature formula for the integral with the implicit trapezoidal method : the trapezoidal rule . One approximates the integrand in every -th step as follows
Together this results in the trapezoidal method
Solution method
Various numerical methods can be used to solve this, usually non-linear, system of equations. For the quadratically convergent Newton method, the following results specifically:
On the imaginary axis , the trapezoid method is A-stable .
Increment
The (variable) step size can be calculated from the following relationship:
;
denotes the permitted local discretization error . The approach provides for the implicit trapezoidal method
.
Here is the absolute value of the largest eigenvalue of the Jacobi matrix ( spectral radius ). The numerical determination of the eigenvalues is very time-consuming; For the purpose of calculating the step size, it is generally sufficient to use the overall standard , which is always greater than or equal to the spectral standard . N is the rank of the Jacobi matrix and its elements.
literature
Hans R. Schwarz, Norbert Köckler: Numerical Mathematics. 5th edition, Teubner, Stuttgart 2004, ISBN 3-519-42960-8 , p. 343.