Optimal Methods

• For nonlinear/nonautonomous systems:
  • Explicit methods
    • Runge-Kutta
    • Multistep
  • Implicit methods
    • Runge-Kutta
    • Multistep

• For linear, autonomous systems:
  • Explicit methods
    • Runge-Kutta
    • Multistep
  • Implicit methods
    • Runge-Kutta
    • Multistep

Matlab Scripts

Maple Scripts

Scripts for Runge-Kutta methods

• Manipulating the coefficient arrays:
  • makebutcher.m - Generates Butcher arrays for various Runge-Kutta methods.
  • butcher2shuosher.m - Generates Shu-Osher arrays for an explicit Runge-Kutta method, given the Butcher array.
  • butcher2modshuosher.m - Generates modified Shu-Osher arrays for a Runge-Kutta method, given the Butcher array.
  • shuosher2butcher.m - Generates Butcher array for a Runge-Kutta method, given the Shu-Osher or modified Shu-Osher arrays.
• Analyzing properties of the methods:
  • rk_order.m - Calculates order of accuracy of a Runge-Kutta method, given the Butcher array.
  • rk_stage_order.m - Calculates order of accuracy of a Runge-Kutta method, given the Butcher array.
  • am_radius.m - Calculates radius of absolute monotonicity of a Runge-Kutta method, given the Butcher array.
  • cc_radius.m - Calculates radius of circle contractivity of a Runge-Kutta method, given the Butcher array.
• Finding optimal methods:
  • Rsp.m - Determines the optimal absolutely monotonic polynomial of degree s and order p.