Low-storage Runge–Kutta methods

LSRK methods are self-starting, with $U^{(1)} = u^n$, and then using stage updates of the form

\[\begin{aligned} dU^{(i)} &= f(U^{(i)}, t + c_i \Delta t) + A_i dU^{(i-1)}\\ U^{(i+1)} &= U^{(i)} + \Delta t B_i dU^{(i)} \end{aligned}\]

where $A_1 = c_1 = 0$ (implying $dU^{(1)} = f(u^n, t)$), with the value at the next step being the $N+1$th stage value $u^{n+1} = U^{(N+1)})$.

This allows the updates to be performed with only two copies of the state vector (so long as f can be evaluated in incrementing form).

It can be written as an RK scheme with Butcher tableau coefficients defined by the recurrences

\[\begin{aligned} a_{i,i} &= 0 \\ a_{i,j} &= B_j + A_{j+1} a_{i,j+1}\\ b_N &= B_N \\ b_i &= B_i + A_{i+1} b_{i+1} \end{aligned}\]

or equivalently

\[\begin{aligned} a_{j,j} &= 0 \\ a_{i,j} &= a_{i-1,j} + B_{i-1} \prod_{k=j+1}^{i-1} A_k \end{aligned}\]

with $b_j$ treated analogously as $a_{N+1,j}$.