Numerical ODE Project List

Tony and Joe's Dreidel project.

Elaheh, Chad, and Jane: Restricted Memory RK Schemes

ref
Low-storage, explicit Runge–Kutta schemes for the compressible
Navier–Stokes equations
Christopher A. Kennedy a;, Mark H. Carpenter b, R. Michael Lewisc
Applied Numerical Mathematics 35 (2000) 177–219

Jason and Rich:  High-Order RK schemes

Nikhil and Chuck.  Customized Explicit RK schemes as "scheme accurate" predictors for implicit schemes.

Robin, Wen, and Wang: Waveform Relaxation:

Stationary Schemes: Low-order composite quadrature schemes (Trap, Mid, Simp, etc.).  Convergence and accuracy.

Stationary Schemes: High-order non-composite quadrature schemes (Newton-Cotes, Gauss-Konrod, Clenshaw-Curtis etc.).  Convergence and accuracy.

Non-stationary schemes interpreted as GLM methods: Variable stepsize schemes, variable advancement, and step-size control.

Starting Methods for WR schemes. Scaling schemes.   {(Subscript[t, 0])+(Subscript[h, 1]),…,(Subscript[t, 0])+(Subscript[h, m])}->{(Subscript[t, 0])+α (Subscript[h, 1]),…,(Subscript[t, 0])+α (Subscript[h, m])} with α>1 for starting methods and α<0 if we need to shrink.

Improving Jacobian recomputation using Broyden like updates.

Taylor/Rosenbrock Like Schemes with Boryden like updates.

Extrapolation Schemes.