18 problems

Let $u^n$ be the numerical solution produced by the nonuniform-time-step GBDF2-IMEX scheme … Here $\tau=\max_{0\leq k\leq M-1}\{\Delta t_k\}$ is the maximum time step, and $h_x$ an…

Sweeping-limiter conjecture. The combination of the robust scaling limiter for pressure correction and the mass-redistribution technique of the sweeping limiter explains why only o…

The linear reaction-convection-diffusion problem is posed on the one-dimensional domain $(0,1)$ with convection coefficient $b(x)=1$, reaction coefficient $c(x)=0$, source term…

A multistep finite difference scheme is understood as a finite difference scheme on the conserved variables arising from a lattice Boltzmann scheme. Uniqueness conjecture. Any mult…

Consider the semi-discrete approximation scheme proposed in the previous work, with parameters satisfying the strict dissipativity condition … Let the SKC condition be fulfilled wh…

Let $M$ be the matrix resulting from the compatibility boundary condition equations on a Cartesian grid with Dirichlet or Neumann boundary conditions, or at a corner where Dirichle…

Consider the energy-conserving Hermite methods for Maxwell's equations with parameter $m$, and let the convergence rate refer to the observed order of convergence in the numerical…

Let $U_{ij}$ be the finite difference solution produced by the two-grid method for the elliptic interface problem … with a linear boundary condition (Dirichlet, Neumann, or Robin),…

The lattice Boltzmann scheme and its corresponding finite difference scheme produce macroscopic equations through their respective equivalent-equation expansions. Equivalence conje…

Let $d$ be a positive integer, and consider a uniform rectangular grid in $d$ dimensions with mesh sizes … Let the grid be shifted along any vector, and apply BFECC interpolation f…

Compact modified-equation stability conjecture. This condition holds at every even order $2p=2,4,6,\ldots$, with a similar result in three-dimensional problems. The condition is kn…

Parameter-scaling conjecture. The relation

Let $N\times N$ be a grid for a finite-difference discretization of the Monge–Ampère Dirichlet problem, and apply Newton's method to the resulting nonlinear system with varying ini…

Let $\theta$ be the MCS parameter, and fix $\lambda=Delta t/h$ and the spatial mesh size $h$. Let the low-wavenumber error be the component denoted by $E_{j,k}^{low}$ in the error…

Let $\theta$ be the MCS parameter, and let the high-wavenumber error be the component denoted by $E_{j,k}^{high}$ in the error decomposition for the Modified Craig–Sneyd scheme. Hi…

Let $\rho$ denote the correlation parameter, $r$ the interest-rate parameter, and let $\mu$ and the remaining model and discretisation parameters take their admissible values. The…

The scheme uses a correlation parameter $\rho$ in the stochastic-volatility model, and its von Neumann stability condition is denoted by … is satisfied also for non-vanishing corre…

The HV1 scheme is an alternating-direction implicit finite-difference scheme for the Heston model, with stability depending on the time-step and spatial discretization through a Co…