= Strong stability preserving Runge-Kutta method
= SSP Runge-Kutta method
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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 $U_{\mathrm{new}}=U+kF(U)+k^2F'(U)F(U)/2+O(k^3)$, 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>.
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