15 problems

Let $L$ be a high-order space discretization of a convective operator, such as an ENO discretization. The fifth-order WENO space discretization is not stable under Forward Euler ti…

Let a vector-valued explicit or implicit Runge–Kutta method have order $p$, and let the conditions in the sufficiency theorem referred to in the source be imposed on such a method.…

Stage-count conjecture. The $R$-th order approximate Taylor method can be cast as a Runge–Kutta method with

Tadmor's strong-stability conjecture. The four-stage Runge–Kutta method fails strong stability: it is not uniformly similar to a contraction. Equivalently, there is no symmetrizer…

Implicitness conjecture. Every symmetric resonance-based scheme is necessarily implicit.

The exact discrete-time model and an approximate discrete-time model are compared through the one-step consistency property REPC, which is sufficient for the multistep consistency…

Automatic commutator-free integrator conjecture. Every $2N$-storage $s$-stage classical Runge–Kutta scheme of order $p$ is also automatically a commutator-free Lie group method of…

Conjecture. At least one of the regions $\mathcal{R}^{MS}$, $\mathcal{R}^{AS}$, and $\mathcal{R}^{SP}$ contains $\mathbb{C}_{-}$.

Let $M_{1,1}$ and $M_{2,2}$ denote the quantities associated with the primary and secondary parts of the additive Runge--Kutta method, and let the primary method have a specified o…

For an explicit third-order SSP Runge–Kutta method, let the $\|\cdot\|_2$-error constant be the constant used to measure its error in the $\|\cdot\|_2$ norm, and let the observed S…

Van der Houwen's conjecture. The optimal stability region touches the circle $C_{m,n}$ at exactly $m-p+2$ distinct points.

Nonexistence conjecture. Within the framework of csRK methods, there exists no computational energy-preserving method that is conjugate to a symplectic method.

Runge–Kutta representation conjecture. The $R$-th-order approximate Taylor method could be cast as a Runge–Kutta method with

Let an $s$-stage fully implicit Runge–Kutta method have order conditions … and let $r>0$ satisfy the absolute-monotonicity conditions … with $I+rA$ invertible and…

Let $m,n\in\mathbb{N}^+$, and let $R_{m/n,2}$ denote the maximal radius of absolute monotonicity among rational functions with numerator degree at most $m$, denominator degree at m…