 Strong stability preserving Runge-Kutta method

A strong stability preserving Runge-Kutta method writes its stages as convex combinations of previously computed states and suitable Forward Euler method steps. It transfers any convex-functional nonincrease property of the forward step, such as a norm bound, under a proportionally scaled time-step restriction.

For example, if $E_k(U)=U+kF(U)$ is nonexpansive in a norm, then $Y=E_k(U)$ and $U_{\mathrm{new}}=(U+E_k(Y))/2$ is also nonexpansive: the triangle inequality bounds the new difference by one half of the initial difference plus one half of the twice-advanced difference. Both are at most the initial difference. A Taylor expansion yields $U_{\mathrm{new}}=U+kF(U)+k^2{F}^{\prime }(U)F(U)/2+O(k^3)$, so this is a second-order Runge-Kutta method. The forward-step hypothesis must hold on the stage states; preservation of one chosen convex bound is not the same as unconditional B-stability.