Two-Step Runge-Kutta Methods

[ChrisRackauckas]

• [ ] http://people.math.sfu.ca/~jverner/TSRKIMPROVEDstart040803.pdf
• [ ] http://people.math.sfu.ca/~jverner/TSRKstart040505.pdf
• [ ] https://arxiv.org/pdf/1106.3626.pdf
• [ ] https://link.springer.com/article/10.1134/S2070048210020092
• [ ] https://www.sciencedirect.com/science/article/pii/S0021999198959861 (one of these have been implemented)

Non-uniform grid: https://link.springer.com/article/10.1023/A:1022352704635

[ranocha]

There are also SSP two-step Runge-Kutta methods:

https://link.springer.com/article/10.1007%2Fs10915-014-9961-7

[kamalojasv181]

@ChrisRackauckas , https://arxiv.org/pdf/1106.3626.pdf says nothing about the stage defining coefficients c. Can u please check

[ChrisRackauckas]

c are just the sum of the row of a.

[kamalojasv181]

Yes but the method is of form this hence I couldn't figure out

[ChrisRackauckas]

it should be the sum of the row of the q's I think? Looks at the first order conditions.

[kamalojasv181]

Yes I thought so. But sum of rows of q =1 for all rows

[ChrisRackauckas]

What's the first order condition here?

[ranocha]

IIRC, you can get the Butcher coefficients c by inserting the ODE u'(t) = 1, u(0) = 0 into the update formula. For the stage values given by (4.1a), you get

time(yᵢⁿ) = dᵢ*time(uⁿ⁻¹) + (1 - dᵢ - ∑ⱼ qᵢⱼ) * time(uⁿ) + ∑ⱼqᵢⱼ * (time(yⱼⁿ) + Δt / r * 1)

Then, the values of c can be computed as

cᵢ = (time(yᵢⁿ) - time(uⁿ)) / Δt

[kamalojasv181]

Thanks. That was helpful.

[kamalojasv181]

https://github.com/SciML/OrdinaryDiffEq.jl/pull/1120