A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains

  title={A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains},
  author={Hans Johansen and Phillip Colella},
  journal={Journal of Computational Physics},
We present a numerical method for solving Poisson's equation, with variable coefficients and Dirichlet boundary conditions, on two-dimensional regions. The approach uses a finite-volume discretization, which embeds the domain in a regular Cartesian grid. We treat the solution as a cell-centered quantity, even when those centers are outside the domain. Cells that contain a portion of the domain boundary use conservative differencing of second-order accurate fluxes on each cell volume. The… 

A Second Order Virtual Node Method for Poisson Interface Problems on Irregular Domains

A discontinuity removal technique that admits use of the standard 5-point finite di erence stencil everywhere in the domain for the specific case of interface problems with continuous coe cients is presented.

An Adaptive Cartesian Grid Embedded Boundary Method for the Incompressible Navier Stokes Equations in Complex Geometry

We present a second-order accurate projection method to solve the incompressible Navier-Stokes equations on irregular domains in two and three dimensions. We use a finite-volume discretization

A finite-difference method for the variable coefficient Poisson equation on hierarchical Cartesian meshes

A Cartesian grid embedded boundary method for the heat equation on irregular domains

An algorithm for solving the heat equation on irregular time-dependent domains is presented, based on the Cartesian grid embedded boundary algorithm of Johansen and Colella, combined with a second-order accurate discretization of the time derivative.

A High Order Cartesian Grid, Finite Volume Method for Elliptic Interface Problems

A simple method based on Green's theorem for computing exact geometric moments directly from an implicit function definition of the embedded interface is developed and produces stencils with a simple bilinear representation.

A Supra-Convergent Finite Difference Scheme for the Poisson and Heat Equations on Irregular Domains and Non-Graded Adaptive Grids

Numerical results in two spatial dimensions demonstrate second order accuracy for both the solution and its gradient in the L 1 and L ∞ norms.



Cartesian grid method for unsteady compressible flow in irregular regions

An adaptive Cartesian grid method for modeling time-dependent inviscid compressible flow in irregular regions using an unsplit second-order Godunov algorithm followed by a corrector applied to cells at the boundary.

Adaptive mesh refinement for hyperbolic partial differential equations

This work presents an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques based upon Richardson-type estimates of the truncation error, which is a mesh refinement algorithm in time and space.

A Cartesian Grid Projection Method for the Incompressible Euler Equations in Complex Geometries

A method for calculating time-dependent incompressible inviscid flow which combines a projection method with a "Cartesian grid" approach for representing geometry, in which the body is represented as an interface embedded in a regular Cartesian mesh.

An adaptive projection method for the incompressible Euler equations

In this paper we present a method for solving the time-dependent incompressible Euler equations on an adaptive grid. The method is based on a projection formulation in which we first solve convection

A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations

A method for solving the equations governing time-dependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids based on a projection formulation in which the first step is to solve advection?diffusion equations to predict intermediate velocities, and then project these Velocities onto a space of approximately divergence-free vector fields.

An Adaptive Mesh Projection Method for Viscous Incompressible Flow

A fractional step version of Chorin's projection method for incompressible flow, with adaptive mesh refinement, which is second-order accurate in both space and time is presented.

A Fast Poisson Solver for Complex Geometries

This paper presents a new fast Poisson solver based on potential theory rather than on direct discretization of the partial differential equation, which combines fast algorithms for computing volume integrals and evaluating layer potentials on a grid with a fast multipole accelerated integral equation solver.

A Multigrid Algorithm for Immersed Interface Problems

This paper describes how to apply the full multigrid algorithm in this context andumerical results are given to demonstrate that good rates can be obtained even when jumps in the coefficients are large and do not align with the grid.

A Projection Method for Locally Refined Grids

The main contributions of this work concern the formulation and implementation of a projection for refined grids, and a method for casting certain approximate projections as MAC projections on refined grids.