Accelerated Runge-Kutta Methods

Figure 1

The central idea of the ARK3 method is illustrated here. Approximating the solution 𝐲_𝑛+1 at 𝑡_𝑛+1 requires 2 function evaluations 𝐤₁ and 𝐤₂ (shown in red) and two reused values 𝐤₋₁ and 𝐤₋₂ (shown in blue) from the previous steps 𝐤₁ and 𝐤₂. This process repeats for the next step (approximating the solution at 𝑡_𝑛+2).