Low-Storage Runge–Kutta Methods

Low-storage Runge–Kutta (LSRK) methods are a class of explicit Runge–Kutta methods designed to minimize memory usage. The $2N$-storage family stores only two state-sized arrays at any time — the current state $U^{(i)}$ and the accumulated increment $dU^{(i)}$ — rather than the $s + 1$ arrays required by a general $s$-stage explicit RK method. The $2N$-storage format was originally introduced by Williamson (1980), and the specific 4th-order families used here were developed by Carpenter and Kennedy (1994) and later extended with optimized stability regions by Niegemann et al. (2012).

Formulation

Each timestep begins with $U^{(1)} = u^n$ and proceeds through $s$ 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 $A_1 = c_1 = 0$ (so the first accumulation reduces to $dU^{(1)} = f(u^n, t)$). The solution at the next timestep is $u^{n+1} = U^{(s+1)}$.

Because each stage only needs $U^{(i)}$ and $dU^{(i-1)}$, the update requires only two state-sized registers (provided $f$ can be evaluated in incrementing form, i.e., f(du, u, p, t, α, β) computes du = α * f(u,p,t) + β * du).

Equivalent Butcher Tableau

The LSRK$(A, B, c)$ scheme can be expressed as a standard explicit Runge–Kutta method with Butcher tableau coefficients defined by the backward recurrence (in $j$):

\[\begin{aligned} a_{i,i} &= 0, \\ a_{i,j} &= B_j + A_{j+1}\, a_{i,j+1} \quad (j < i),\\ b_i &= B_i + A_{i+1}\, b_{i+1},\quad b_s = B_s, \end{aligned}\]

or equivalently by the forward recurrence (in $i$):

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

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

Implemented Algorithms

The following LSRK algorithms are implemented in ClimaTimeSteppers.jl as subtypes of LowStorageRungeKutta2N:

LSRK54CarpenterKennedy is a compact, widely used 4th-order method with only 5 stages. LSRK144NiegemannDiehlBusch uses 14 stages to achieve a much larger explicit stability region, making it well suited for hyperbolic problems where the maximum stable timestep is the limiting factor.

Usage with Multirate Methods

The LowStorageRungeKutta2N family can also serve as the outer solver in a multirate Runge–Kutta scheme (see Multirate Runge–Kutta). The incremental form of each stage maps naturally onto the slow-tendency accumulation needed by the MRRK framework.