Time-dependent boundary and interior forcing in locally one-dimensional schemes
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Numerical Analysis
Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
A Cartesian grid embedded boundary method for Poisson's equation on irregular domains
Journal of Computational Physics
Computation of solid-liquid phase fronts in the sharp interface limit on fixed grids
Journal of Computational Physics
SIAM Journal on Numerical Analysis
The immersed interface method for the Navier-Stokes equations with singular forces
Journal of Computational Physics
A Cartesian grid embedded boundary method for the heat equation on irregular domains
Journal of Computational Physics
Immersed Interface Methods for Neumann and Related Problems in Two and Three Dimensions
SIAM Journal on Scientific Computing
An Immersed Interface Method for Incompressible Navier-Stokes Equations
SIAM Journal on Scientific Computing
A ghost-cell immersed boundary method for flow in complex geometry
Journal of Computational Physics
A node-centered local refinement algorithm for Poisson's equation in complex geometries
Journal of Computational Physics
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Journal of Computational Physics
An immersed interface method for simulating the interaction of a fluid with moving boundaries
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
A new interface tracking method: The polygonal area mapping method
Journal of Computational Physics
Journal of Computational Physics
HyPAM: A hybrid continuum-particle model for incompressible free-surface flows
Journal of Computational Physics
SIAM Journal on Scientific Computing
Journal of Scientific Computing
Hi-index | 31.45 |
The ghost cell approaches (GCA) for handling stationary solid boundaries, regular or irregular, are first investigated theoretically and numerically for the diffusion equation with Dirichlet boundary conditions. The main conclusion of this part of investigation is that the approximation for the diffusion term has to be second-order accurate everywhere in order for the numerical solution to be rigorously second-order accurate. Violating this principle, the linear and quadratic GCAs have the following shortcomings: (1) restrictive constraints on grid size when the viscosity is small; (2) susceptibleness to instability of a time-explicit formulation for strongly transient flows; (3) convergence deterioration to zeroth- or first-order for solutions with high-frequency modes. Therefore, the widely-used linear extrapolation for enforcing no-slip boundary conditions should be avoided, even for regular solid boundaries. As a remedy, a simple method based on explicit jump approximation (EJA) is proposed. EJA hinges on the idea that a velocity-derivative jump at the boundary reduces to the value of the velocity-derivative at the fluid side because the velocity of the stationary boundary is zero. Although the time-marching linear system of EJA is not symmetric, it is strictly diagonal dominant with positive diagonal entries. Numerical results show that, over a large range of viscosity and grid sizes, EJA performs much better than GCAs in terms of stability and accuracy. Furthermore, the second-order convergence of EJA does not depend on viscosity and the spectrum of the solution, as those of GCAs do. This paper is written with enough details so that one can reproduce the numerical results.