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#### Finite volume methods and adaptive refinement for global tsunami propagation and local inundation

(2006)

The shallow water equations are a commonly accepted approximation governing tsunami propagation. Numerically capturing certain features of local tsunami inundation requires solving these equations in their physically relevant conservative form, as integral conservation laws for depth and momentum. This form of the equations ...

#### High-resolution rotated grid method for conservation laws with embedded geometries

(2005)

We develop a second-order rotated grid
method for the approximation of time dependent solutions of conservation laws in complex
geometry using an underlying Cartesian grid. Stability for time steps adequate for the
regular part of the grid is obtained by increasing the domain of dependence of the numerical
method ...

#### High-resolution finite volume methods for dusty gas jets and plumes

(2006)

We consider a model for dusty gas flow that consists of the compressible Euler equations for the gas coupled to a similar (but pressureless) system of equations for the mass, momentum, and energy of the dust. These sets of equations are coupled via drag terms and heat transfer. A high-resolution wave-propagation algorithm is ...

#### CORRECTION TO THE ARTICLE A COMPARISON OF THE EXTENDED FINITE ELEMENT METHOD WITH THE IMMERSED INTERFACE METHOD FOR ELLIPTIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS AND SINGULAR SOURCES BY VAUGHAN ET AL.

(2008)

A recent paper
by Vaughan, Smith, and Chopp [Comm. App. Math. &and Comp. Sci.
1 (2006), 207–228] reported numerical results for three
examples using the immersed interface method (IIM) and the extended
finite element method (X-FEM). The results presented for the IIM
showed first-order accuracy for the solution and ...

#### A wave propagation method for three-dimensional hyperbolic conservation laws

(Elsevier, 2000-11-20)

A class of wave propagation algorithms for three-dimensional conservation laws and other hyperbolic systems is developed. These unsplit finite-volume methods are based on solving one-dimensional Riemann problems at the cell interfaces and applying flux-limiter functions to suppress oscillations arising from second-derivative ...

#### A class of approximate Riemann Solvers and their relation to relaxation schemes

(Elsevier/Academic Press, 2001-09-30)

We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math.48, 235 (1995)] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed ...

#### A wave propagation algorithm for hyperbolic systems on curved manifolds

(Elsevier, 2004-09-20)

An extension of the wave propagation algorithm first introduced by LeVeque [J. Comput. Phys. 131 (1997) 327] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, shallow water flow on the surface ...

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

(Society for Industrial and Applied Mathematics, 2002)

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 ...