Trapezoidal method

The implicit trapezoidal method is a method for numerically solving an initial value problem

${\displaystyle y'(t)=f\left(t,y(t)\right),\quad y(t_{0})=y_{0}}$

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 : ${\displaystyle y'=\mathrm {i} \alpha y}$

${\displaystyle y_{n+1}=y_{n}+{\frac {h}{2}}(f_{n+1}+f_{n})}$

With

${\displaystyle f_{n}\ :=f(t_{n},y_{n}).}$

Derivation

For the derivation of one-step procedure which is an initial value problem mostly in the equivalent to her integral equation reshaped

${\displaystyle {\begin{aligned}{\dot {y}}&=f(t,y)\,,\qquad y(t_{0})=y_{0}\\\Longleftrightarrow \quad y(t)&=y_{0}+\int _{t_{0}}^{t}f(s,y(s))\,\mathrm {d} s\,.\end{aligned}}}$

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 ${\displaystyle k}$

${\displaystyle \int _{t_{k}}^{t_{k+1}}f(s,y(s))\,\mathrm {d} s\approx {\frac {h}{2}}{\Big (}f(t_{k},y_{k})+f(t_{k+1},y_{k+1}){\Big )}\,.}$

Together this results in the trapezoidal method

${\displaystyle y(t_{k+1})=y(t_{k})+\int _{t_{k}}^{t_{k+1}}f(s,y(s))\,\mathrm {d} s\approx y_{k}+{\frac {h}{2}}{\Big (}f(t_{k},y_{k})+f(t_{k+1},y_{k+1}){\Big )}=:y_{k+1}\,.}$

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:

${\displaystyle y_{n+1}^{(k+1)}=y_{n+1}^{(k)}-\left(I-{\frac {h}{2}}{\frac {\partial f_{n+1}^{(k)}}{\partial y_{n+1}^{(k)}}}\right)^{-1}\left(y_{n+1}^{(k)}-y_{n}-{\frac {h}{2}}(f_{n+1}^{(k)}+f_{n})\right).}$

So you get a linear system of equations

${\displaystyle (I-{\frac {h}{2}}J^{(k)})y_{n+1}^{(k+1)}=-{\frac {h}{2}}J^{(k)}y_{n+1}^{(k)}+y_{n}+{\frac {h}{2}}(f_{n+1}^{(k)}+f_{n}),}$

where J is the Jacobian matrix

${\displaystyle J^{(k)}:=\left({\frac {\partial f}{\partial y}}\right)_{n+1}^{(k)}}$,

${\displaystyle I}$is the identity matrix and the iteration step. ${\displaystyle k}$

stability

The test equation gives the stability function ${\displaystyle y'(t)=\lambda y(t)}$

${\displaystyle R(z)={\frac {2+z}{2-z}},\quad z=h\lambda \in \mathbb {C} .}$

On the imaginary axis , the trapezoid method is A-stable . ${\displaystyle z=i\eta }$ ${\displaystyle |R(i\eta )|=1}$

Increment ${\displaystyle h}$

The (variable) step size can be calculated from the following relationship:

${\displaystyle \left\vert {\frac {R(h\lambda )}{e^{h\lambda }}}-1\right\vert =\delta }$;

${\displaystyle \delta }$denotes the permitted local discretization error . The approach provides for the implicit trapezoidal method ${\displaystyle y_{n+1}=y_{n}+{\frac {h}{2}}(f_{n+1}+f_{n})=:R(h\lambda )y_{n}}$

${\displaystyle R(h\lambda )={\frac {2+h\lambda }{2-h\lambda }}}$.

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. ${\displaystyle \lambda \,:=\max _{j}{|\lambda _{j}|}}$ ${\displaystyle \lambda =N\cdot \max _{i,j}|a_{ij}|}$ ${\displaystyle a_{ij}}$

literature

• Hans R. Schwarz, Norbert Köckler: Numerical Mathematics. 5th edition, Teubner, Stuttgart 2004, ISBN 3-519-42960-8 , p. 343.

Individual evidence

1. ↑ M. Kloker: Numerical solvers (time integration method ) for the common model differential equation y '= αy (PDF; 2.2 MB), University of Stuttgart, 1996
2. ↑ Reusken, Arnold .: numerics for engineers and scientists . Springer, Berlin 2006, ISBN 3-540-25544-3 , pp. 378 .
3. ↑ Reusken, Arnold .: numerics for engineers and scientists . Springer, Berlin 2006, ISBN 3-540-25544-3 , pp. 383 .