Dynamic errors of numerical methods of ODE discretization

The dynamic characteristic of the numerical method of ordinary differential equations (ODE) discretization is the natural logarithm of its stability function $\text{D}=\ln \rho (h\lambda )$. Dynamic characteristics are considered in three forms:

$\text{D}$ – Complex dynamic characteristic;

${\text{D}}_R$ – Real dynamic characteristic;

${\text{D}}_I$ – Imaginary dynamic characteristic.

The dynamic characteristic represents the transformation operator of eigenvalues of a Jacobian matrix of the initial differential mathematical model (MM) into eigenvalues of a Jacobian matrix of a mathematical model (also differential) whose exact solution passes through the discrete sequence of points of the initial MM solution obtained by the given numerical method.

See also

• Euler's method
• Runge–Kutta methods
• Runge–Kutta method (SDE)
• List of Runge–Kutta methods

References

1. Kosteltsev V.I. Dynamic properties of numerical methods of integration of systems of ordinary differential equations. – Preprint N23. – L.: LIIAN, 1986.
2. Dekker K., Verver J. Stability of Runge–Kutta methods for stiff nonlinear differential equations. / trans. from engl. – M.: Mir, 1988.

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