17 problems

On the unit grid, let $\bar u=(\bar u_i)_{i\in\mathbb Z}$ be a compactly supported sequence of cell averages, define the forward difference by $(\mathrm Db)_i:=b_{i+1}-b_i$, set…

A numerical method for the Euler equations is considered, with vorticity as a diagnostic of anomalous numerical solutions such as carbuncle phenomena and unphysical Mach stems. Mor…

Fully discrete methods use a compact stencil in both space and time when each update depends only on a bounded local set of spatial and temporal degrees of freedom. Roe et al.'s co…

The hybrid pressure formulation produces a global problem matrix obtained from the discretization of viscous laminar incompressible flows. Negative-definiteness conjecture. The mat…

Let $\Omega$ be the computational domain, let $\Gamma_- = \{\mathbf x\in\Gamma: \mathbf v(\mathbf x)\cdot\mathbf n(\mathbf x)<0\}$ be the inflow part of the boundary, let…

Let $h_{\mathcal T}$ denote the mesh size, let $\overline{u}$ be the exact solution, let $\overline{u}_{\mathcal T}$ be its interpolation in $X_{\mathcal T}$, and let…

The paper considers piecewise linear reconstructions of cell-averaged data and the graphical slope-limiter rule that bounds the reconstructed slope by four lines connecting neighbo…

For a monotone finite volume scheme approximating a scalar conservation law with initial data of Hölder regularity $balpha$, let {bDelta x}} denote the spatial resolution. Optima…

Let $u^0$ be a function with total bounded variation, and associate to it the initialization … Suppose that the ratio $\lambda=\frac{\Delta t}{V\Delta x}$ is fixed and belongs to…

Let $u_0$ be initial data such that $H(u_0)\belongs to L^1(\Omega)$, and let $u$ be a weak solution in the sense of Definition. Renormalization conjecture. The weak solution $u$ is…

The scheme approximates a density $\rho$ for a vehicular-flow initial-boundary value problem, with entropy solutions understood through the boundary trace condition associated with…

Let $\overline{v}=(\overline{v}_i)_{i\in\mathbb{Z}}$ be cell averages, let $\llbracket\overline{v}\rrbracket_{i+\frac{1}{2}}$ denote the reconstructed jump at the interface…

Let $k$ denote the order of the ENO reconstruction method. The reconstructed TV bound is the bound referred to as, and the weak TV bound is the estimate … for some $p\geqslant1$ an…

Convergence-loss conjecture. This mismatch between the linearised wave cones and the physical domain of dependence is the origin of the loss of convergence for Thacker's planar sol…

The proposed hybrid finite difference–finite volume schemes evolve both cell-averaged and nodal values for first-order hyperbolic conservation laws, combining conservation with hig…

The MinMod scheme has a similarity solution whose higher derivatives may be discontinuous. Slow-convergence conjecture. These discontinuities result in slow convergence. The conjec…

The schemes use weights $w_j$ to satisfy the averaging conditions for reconstructing values at vertices from surrounding tetrahedral cell centers. The modified Shepard interpolatio…