Randall J. LeVeque

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources

The authors develop finite difference methods for elliptic equations of the form \[ \nabla \cdot (\beta (x)\nabla u(x)) + \kappa (x)u(x) = f(x)\] in a region

High-resolution conservative algorithms for advection in incompressible flow

\Omega

Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension

in one or two space dimension...

A study of numerical methods for hyperbolic conservation laws with stiff source terms

A class of high-resolution algorithms is developed for advection of a scalar quantity in a given incompressible flow field in one, two, or three space dimensions. Multidimensional transport is modeled using a wave-propagation approach in which the flux at each cell interface is built up on the basis of information propagating in the direction of this interface from neighboring cells. A high-resolution second-order method using slope limiters is quite easy to implement. For constant flow, a minor modification gives a third-order accurate method. These methods are stable for Courant numbers up to 1. Fortran implementations are available by anonymous ftp.

An Immersed Interface Method for Incompressible Navier-Stokes Equations

A second-order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the second-order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper [SIAM J. Numer. Anal., 31(1994), pp. 1019--1044]. The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasi-Newton method is developed that allows reasonable time steps to be used.

A Wave Propagation Method for Conservation Laws and Balance Laws with Spatially Varying Flux Functions

The proper modeling of nonequilibrium gas dynamics is required in certain regimes of hypersonic flow. For inviscid flow this gives a system of conservation laws coupled with source terms representing the chemistry. Often a wide range of time scales is present in the problem, leading to numerical difficulties as in stiff systems of ordinary differential equations. Stability can be achieved by using implicit methods, but other numerical difficulties are observed. The behavior of typical numerical methods on a simple advection equation with a parameter-dependent source term was studied. Two approaches to incorporate the source term were utilized: MacCormack type predictor-corrector methods with flux limiters, and splitting methods in which the fluid dynamics and chemistry are handled in separate steps. Various comparisons over a wide range of parameter values were made. In the stiff case where the solution contains discontinuities, incorrect numerical propagation speeds are observed with all of the methods considered. This phenomenon is studied and explained.

Adaptive Mesh Refinement Using Wave-Propagation Algorithms for Hyperbolic Systems

The method developed in this paper is motivated by Peskins immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points. Our method instead incorporates part of this force into jump conditions for the pressure, avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary. This has been implemented for the two-dimensional incompressible Navier--Stokes equations using a high-resolution finite-volume method for the advective terms and a projection method to enforce incompressibility. In the projection step, the correct jump in pressure is imposed in the course of solving the Poisson problem. This gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional IB method. Comparisons between this method and the IB method are presented for several test problems. Numerical studies of the convergence and order of accuracy are included.

Algorithms for Computing the Sample Variance: Analysis and Recommendations

We study a general approach to solving conservation laws of the form qt+f(q,x)x=0, where the flux function f(q,x) has explicit spatial variation. Finite-volume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a generalized Riemann problem between neighboring grid cells. A high-resolution wave-propagation algorithm is defined in which waves are based directly on a decomposition of flux differences fi(Qi)-f-1(Qi-1) into eigenvectors of an approximate Jacobian matrix. This method is shown to be second-order accurate for smooth problems and allows the application of wave limiters to obtain sharp results on discontinuities. Balance laws

Analysis of a one-dimensional model for the immersed boundary method

q_t+f(q,x)_x=\psi(q,x)

An Adaptive Cartesian Mesh Algorithm for the Euler Equations in Arbitrary Geometries

are also considered, in which case the source term is used to modify the flux difference before performing the wave decomposition, and an additional term is derived that must also be included to obtain full accuracy. This method is particularly useful for quasi-steady problems close to steady state.