## A wave-propagation method for conservation laws and balance laws with spatially varying flux functions (2002)

Venue: | SIAM J. Sci. Comput |

Citations: | 29 - 5 self |

### BibTeX

@ARTICLE{Bale02awave-propagation,

author = {Derek S. Bale and Randall J. Leveque and Sorin Mitran and James A. Rossmanith},

title = {A wave-propagation method for conservation laws and balance laws with spatially varying flux functions},

journal = {SIAM J. Sci. Comput},

year = {2002},

volume = {24},

pages = {2002}

}

### OpenURL

### Abstract

Abstract. 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)− fi−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 qt + f(q, x)x = ψ(q, x) 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. Key words. finite-volume methods, high-resolution methods, conservation laws, source terms, discontinuous flux functions AMS subject classifications. 65M06, 35L65 PII. S106482750139738X

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Citation Context ... is not a reasonable model of reality. To more properly model this situation we should use a nonlinear traffic model in which the velocity depends on the density, for example, the classical flux (see =-=[19]-=-, [31]) (2.7) fi(q)=ui(1 − q)q, where ui is now the maximum speed (at q = 0)but the speed drops linearly to 0 as the density increases to q = 1 (bumper-to-bumper traffic). In this case even taking ui−... |

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Citation Context ...(q)x =0, where f depends only on q, has been studied extensively, and a variety of numerical methods have been developed. We concentrate on extending the high-resolution wavepropagation algorithms of =-=[20]-=- to the case of a spatially varying flux function. These are finite-volume methods in which a cell average (1.3) Q n i ≈ 1 ∆x � xi+1/2 x i−1/2 q(x, tn) dx is updated in each time step. This is done us... |

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Citation Context .../2 . p=1 Now � Zp is a limited version of the f-wave Zp obtained in the same manner as � Wp would be obtained from Wp . The standard wave-propagation algorithm is implemented in the Clawpack software =-=[18]-=-, and this can be applied to the problems considered here by defining Wp = Zp /sp , since we assume sp �= 0. However, the modification just described appears more robust since we do not need to worry ... |