Integreat`RK`
Integreat`RK`

RKLinearStability

RKLinearStability[rk, z]

evaluates the linear stability function of rk at z.

Details and Options

  • The linear stability function of a Runge–Kutta method is , where is a vector of 's.
  • When a Runge–Kutta method is applied to the Dahlquist test problem, , the solution is , where .
  • If z is a DirectedInfinity, then RKLinearStability[rk,z] evaluates the stability function in the limit. This is useful for checking L-stability.
  • The following options can be given:
  • EmbeddedFalsewhether to use the embedded coefficients
    StageNonetreat a stage as the solution
    DenseOutputFalsehow to evaluate dense output
  • RKLinearStability[rk,z,StageAll] gives the internal stability function of rk: .

Examples

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Basic Examples  (1)

The linear stability function of the classical fourth order Runge–Kutta method:

Scope  (1)

The evaluation point may be infinite:

Options  (3)

Embedded  (1)

Get the linear stability function for an embedded method:

Stage  (1)

Evaluate the internal stability function:

Evaluate only for stage two:

DenseOutput  (1)

Get the linear stability function for the dense output solution:

Tech Notes
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  • Runge–Kutta Methods