## Numerical Techniques for Conservation Laws with Source Terms, MSc Dissertation (1998)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Hudson98numericaltechniques,

author = {Justin Hudson and Prof M. J. Baines},

title = {Numerical Techniques for Conservation Laws with Source Terms, MSc Dissertation},

year = {1998}

}

### OpenURL

### Abstract

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

### Citations

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Citation Context ...] v ui − ui − Ri + Ri+ 1 if v < 0 ⎩ 2 Unfortunately, this scheme is only a first order approximation to (4.10) but we can also obtain a second order accurate scheme by using van Leer’s MUSCL approach =-=[10]-=- to obtain u n = ui − v ui v 2 n n n n n n n ( − ) − ( 1− v)[ − + ]+ Δt[ ( − α) + ] n+ 1 i i−1 i+ 1 i i−1 1 i α i−1 where c > 0. And hence we may obtain u u ⎧ n n v n n n n n ⎪ v( ui − ui−1) + ( 1− v)... |

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Citation Context ...herefore second order accurate and Figure 2-6 shows that the Lax-Wendroff scheme with Superbee flux-limiter method is considerably more accurate than without the Superbee fluxlimiter method. See Sweby=-=[12]-=- for a more in-depth analysis on flux-limiter methods. 30φ(θ) 3 φ(θ) = θ 2 1 φ(θ) = 1 1 2 3 θ Figure 2-7: TVD region for finite difference schemes. φ(θ) 3 φ(θ) = θ 2 1 φ(θ) = 1 1 2 3 θ Figure 2-8: Se... |

161 |
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Citation Context ... a 11great deal more and definitely too many to look at in this section. For a more in depth discussion of finite difference schemes for the advection equation, look in Kroner[8], LeVeque[7] and Ames=-=[14]-=-. 2.2 1-D Conservation Law In Section 2.1, we discussed some finite difference schemes for approximating the linear advection equation, which is a form of the scalar conservation law ∂u ∂f ( u) + = 0 ... |

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Citation Context ...s well as x and t, resulting in the numerical approximation of the source term no longer being exact. Throughout this chapter, we will be using the following test problem considered by LeVeque and Yee=-=[1]-=-. where with initial data ∂u ∂u + = ∂t ∂x R() u ⎛ 1 ⎞ R , u () u = −u( u −1) ⎜u − ⎟ ⎝ 2 ⎠ 1 ( ) ⎨ ⎩ ⎧ x, 0 = 0 if if x ≤ x > , (4.2) 0. 3 0. 3 and whose exact solution is to illustrate some numerical ... |

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Citation Context ...odels flow in rivers for a channel of finite depth and requires the numerical solution of a system of equations of the form (1.1). Consider the Shallow Water Equation discussed by Bermudez and Vazques=-=[4]-=- where ∂w ∂F + ∂t ∂x ⎡h⎤ ⎡ h ⎤ ⎡ w ( x, t) = ⎢ ⎥ = ⎢ ⎥ , ( w) = ⎢q ⎣q⎦ ⎣uh⎦ ⎢ ⎣ h ( w) = R q ⎤ ⎥ + gh ⎥ 2 ⎦ 2 F 2 ( x, w) 1 and R( x w) () ⎥ ⎡ 0 ⎤ , = ⎢ . ⎣ghH ′ x ⎦ (1.2) Here, h(x,t) and u(x,t) repr... |

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Citation Context ...st order schemes suffer from dissipation but are non-dispersive, and all second order schemes suffer from dispersion but are non-dissipative. For a more in depth discussion on wave theory, see Whitham=-=[9]-=- and Ames[14]. 2.5 Flux-limiter Methods So far we have seen that, in general, all first order schemes suffer from dissipation and all second order schemes suffer from dispersion, which creates oscilla... |

22 |
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Citation Context ...ut there are a 11great deal more and definitely too many to look at in this section. For a more in depth discussion of finite difference schemes for the advection equation, look in Kroner[8], LeVeque=-=[7]-=- and Ames[14]. 2.2 1-D Conservation Law In Section 2.1, we discussed some finite difference schemes for approximating the linear advection equation, which is a form of the scalar conservation law ∂u ∂... |

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Citation Context ... 0.5 0.6 0.7 0.8 0.9 1 -0.2 x Exact MPDATA MPDATA + TVD Figure 4-3: MPDATA approach with and without Superbee flux-limiter. 4.2 Roe’s Upwind Approach 4.2.1 Advection Equation with Source Term R(x) Roe=-=[6]-=- derived a finite difference scheme which numerically approximates ∂u ∂u + c ∂t ∂x = R(x) , (4.10) where c > 0 and R(x) is the source term, with second order accuracy. If we consider the initial-value... |

9 |
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Citation Context ...arkiewicz + Margolin[3] Roe’s Explicit Roe[6], (4.21) 1 Upwind I Vazquez + Bermudez[4] Roe’s Explicit Roe[6], (4.23) 2 Upwind II Vazquez + Bermudez[4] Implicit Upwind I (4.30) 1 Embid, Goodman + Majda=-=[2]-=- Implicit Upwind II (4.34) 2 Embid, Goodman + Majda[2] Yee[5], Explicit (4.37) 2 LeVeque + Yee[1], MacCormack Embid, Goodman + Majda[2] Semi-Implicit Yee[5], (4.39) 2 MacCormack LeVeque + Yee[1] Split... |

9 |
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Citation Context ...n equation but there are a 11great deal more and definitely too many to look at in this section. For a more in depth discussion of finite difference schemes for the advection equation, look in Kroner=-=[8]-=-, LeVeque[7] and Ames[14]. 2.2 1-D Conservation Law In Section 2.1, we discussed some finite difference schemes for approximating the linear advection equation, which is a form of the scalar conservat... |

8 |
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(Show Context)
Citation Context ...lso a function of u, the approximations of the source term are not as accurate making the two schemes accuracy change dramatically, as we will see later. 3.3 MPDATA approach Smolarkiewicz and Margolin=-=[3]-=- derived an algorithm to approximate the advection transport equation (3.7) called MPDATA. MPDATA is a Multidimensional Positive Definite Advection Transport Algorithm and approximates the advection e... |

8 | C.: Upwind and Symmetric Shock-Capturing Schemes - Yee - 1987 |

1 |
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- Sweby
(Show Context)
Citation Context ...x Exact Upwind Figure 2-4: Dissipation of the first order Upwind scheme In order fully understand why dissipation occurs, we will use the analysis of the modified equation, which is discussed by Sweby=-=[13]-=- and LeVeque[7], on the LaxFriedrichs scheme for the advection equation u 1 n n v n n = ( ui+ 1 + ui−1) − [ ui+ 1 ui− ]. 2 2 n+ 1 i − 1 22Earlier, we saw that this scheme had a truncation error of Τ ... |