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Balancing Source Terms and Flux Gradients in HighResolution Godunov Methods: The QuasiSteady WavePropogation Algorithm
 J. Comput. Phys
, 1998
"... . Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of suc ..."
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Cited by 54 (5 self)
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. Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of such states. Here a variant of the wavepropagation algorithm is developed which addresses this problem by introducing a Riemann problem in the center of each grid cell whose flux difference exactly cancels the source term. This leads to modified Riemann problems at the cell edges in which the jump now corresponds to perturbations from the steady state. Computing waves and limiters based on the solution to these Riemann problems gives highresolution results. The 1D and 2D shallow water equations for flow over arbitrary bottom topography are use as an example, though the ideas apply to many other systems. The method is easily implemented in the software package clawpack. Keywords: Godunov meth...
A Wave Propagation Method for Three Dimensional Hyperbolic Problems
, 1996
"... A class of wave propagation algorithms for threedimensional conservation laws is developed. This unsplit nite volume method is based on solving onedimensional Riemann problems at the cell interfaces and applying fluxlimiter functions to suppress oscillations arising from second derivative terms. ..."
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Cited by 23 (5 self)
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A class of wave propagation algorithms for threedimensional conservation laws is developed. This unsplit nite volume method is based on solving onedimensional Riemann problems at the cell interfaces and applying fluxlimiter functions to suppress oscillations arising from second derivative terms. Waves emanating from the Riemann problem are further split by solving Riemann problems in the transverse direction to model crossderivative terms. Due to proper upwinding, the method is stable for Courant numbers up to one. Several examples using the Euler equations are included.
Shock capturing and front tracking methods for granular avalanches
 J. Comput. Phys
, 2002
"... Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shockcapturing numerical scheme for the onedimensional Savage–Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme ..."
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Cited by 10 (4 self)
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Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shockcapturing numerical scheme for the onedimensional Savage–Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme applied to the nonconservative form of the equations reproduces smooth solutions of these free boundary problems very well, but fails when shocks are formed. A nonoscillatory central (NOC) difference scheme with TVD limiter or WENO cell reconstruction for the conservative equations is therefore introduced. For the avalanche free boundary problems it must be combined with a fronttracking method, developed here, to properly describe the margin evolution. It is found that this NOC scheme combined with the fronttracking module reproduces both the shock wave and the smooth solution accurately. A piecewise quadratic WENO reconstruction improves the smoothness of the solution near local extrema. The schemes are checked against exact solutions for (1) an upward moving shock wave, (2) the motion of a parabolic cap down an inclined plane, and (3) the motion of a parabolic cap down a curved slope ending in a flat runout region, where a shock is formed as the avalanche comes to a halt. c ○ 2002 Elsevier Science Key Words: granular avalanche; shockcapturing; nonoscillatory central scheme; free moving boundary; fronttracking. 1.
Nonlinear Conservation Laws and Finite Volume Methods for Astrophysical Fluid Flow
 Computational Methods for Astrophysical Fluid Flow, 27th SaasFee Advanced Course Lecture Notes
, 1998
"... Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1 Software : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : ..."
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Cited by 6 (0 self)
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Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1 Software : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.3 Classification of differential equations : : : : : : : : : : : : : : : : : : : : : : : 7 2. Derivation of conservation laws : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.1 The Euler equations of gas dynamics : : : : : : : : : : : : : : : : : : : : : : : 13 2.2 Dissipative fluxes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.3 Source terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.4 Radiative trans
Numerical Techniques for Conservation Laws with Source Terms, MSc Dissertation
, 1998
"... In this dissertation we will discuss the finite difference method for approximating conservation laws with a source term present which is considered to be a known function of x, t and u. Finite difference schemes for approximating conservation laws without a source term present are discussed and are ..."
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In this dissertation we will discuss the finite difference method for approximating conservation laws with a source term present which is considered to be a known function of x, t and u. Finite difference schemes for approximating conservation laws without a source term present are discussed and are adapted to approximate conservation laws with a source term present. First we consider the source term to be a function of x and t only and then we consider the source term to be a function of u also. Some numerical results of the different approaches are discussed throughout the dissertation and an overall comparison of
NUMERICAL CONSERVATION PROPERTIES OF H(div)CONFORMING LEASTSQUARES FINITE ELEMENT METHODS FOR THE BURGERS EQUATION ∗
"... Abstract. Leastsquares finite element methods (LSFEMs) for the inviscid Burgers equation are studied. The scalar nonlinear hyperbolic conservation law is reformulated by introducing the flux vector, or the associated flux potential, explicitly as additional dependent variables. This reformulation h ..."
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Abstract. Leastsquares finite element methods (LSFEMs) for the inviscid Burgers equation are studied. The scalar nonlinear hyperbolic conservation law is reformulated by introducing the flux vector, or the associated flux potential, explicitly as additional dependent variables. This reformulation highlights the smoothness of the flux vector for weak solutions, namely, f(u) ∈ H(div, Ω). The standard leastsquares (LS) finite element (FE) procedure is applied to the reformulated equations using H(div)conforming FE spaces and a Gauss–Newton nonlinear solution technique. Numerical results are presented for the onedimensional Burgers equation on adaptively refined spacetime domains, indicating that the H(div)conforming FE methods converge to the entropy weak solution of the conservation law. The H(div)conforming LSFEMs do not satisfy a discrete exact conservation property in the sense of Lax and Wendroff. However, weak conservation theorems that are analogous to the Lax–Wendroff theorem for conservative finite difference methods are proved for the H(div)conforming LSFEMs. These results illustrate that discrete exact conservation in the sense of Lax and Wendroff is not a necessary condition for numerical conservation but can be replaced by minimization in a suitable continuous norm. Key words. leastsquares variational formulation, finite element discretization, Burgers equation, nonlinear hyperbolic conservation laws, weak solutions
unknown title
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
Version 1.2 Numerical Techniques for the Shallow Water Equations
"... In this report we will discuss some numerical techniques for approximating the Shallow Water equations. In particular we will discuss finite difference schemes, adaptations of Roe’s approximate Riemann solver and the QSchemes of Bermudez & Vazquez with the objective of accurately approximating the ..."
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In this report we will discuss some numerical techniques for approximating the Shallow Water equations. In particular we will discuss finite difference schemes, adaptations of Roe’s approximate Riemann solver and the QSchemes of Bermudez & Vazquez with the objective of accurately approximating the solution of the Shallow Water equations. We consider four different test problems for the Shallow Water equations with each test problem making the source term more significant, i.e. the variation of the Riverbed becomes more pronounced, so that the different approaches discussed in this report can be rigorously tested. A comparison of the different approaches discussed in this report will also be made so that we may determine which approach produced the most accurate numerical results overall.
WellBalanced Schemes for the Initial Boundary Value Problem for 1D Scalar Conservation Laws
, 2006
"... We consider wellbalanced numerical schemes for the following 1D scalar conservation law with source term: ∂tu + ∂xf (u) + z ′ (x) b(u) = 0. More precisely, we are interested in the numerical approximation of the initial boundary value problem for this equation. While our main concern is a converge ..."
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We consider wellbalanced numerical schemes for the following 1D scalar conservation law with source term: ∂tu + ∂xf (u) + z ′ (x) b(u) = 0. More precisely, we are interested in the numerical approximation of the initial boundary value problem for this equation. While our main concern is a convergence result, we also have to extend Otto’s notion of entropy solutions to conservation laws with a source term. To obtain uniqueness, we show that a generalization, the socalled entropy process solution (see [7]), is unique and coincides with the entropy solution. If the initial and boundary data are in L ∞ , we can establish convergence to the entropy solution. Showing that the numerical solutions are bounded we can extract a weak∗convergent subsequence. Identifying its limit as an entropy process solution requires some effort as we cannot use Kruˇzkovtype entropy pairs here. We restrict ourselves to the EngquistOsher flux and identify the numerical entropy flux for an arbitrary entropy pair. By the uniqueness result, the scheme then approximates the entropy solution and a result by Vovelle then guarantees that the convergence is strong in Lp, 1 ≤ p < ∞. 1 1