Finite Volume Methods for Nonlinear Scalar Conservation Laws
We now turn to the development of finite volume methods for nonlinear conservation laws. We will build directly on what has already been developed for linear systems in Chapter 4, concentrating in this chapter on scalar equations, although much of what is developed will also apply to nonlinear systems of equations. In Chapter 15 we consider some additional issues arising for systems of equations, particularly the need for efficient approximate Riemann solvers in Section 15.3. Nonlinearity...
ul ur
and so the entropy condition is satisfied only if ul gt ur, as we already found from the Lax Entropy Condition 11.1. In general an entropy function should be strictly convex, with a second derivative that is strictly positive at every point. This is crucial in the above analysis of 11.50 , since the qx 2 term that gives rise to the inequality in 11.51 is multiplied by n qe . Rather than considering a single strictly convex function that can be used to investigate every value of qe, a different...
Info Hjk
for all values of 0 gt 0 in order to guarantee that condition 6.44 is satisfied along with 0 0 0 for 0 lt 0 . These constraints can be rewritten concisely as 0 lt 0 0 lt minmod 2, 20 . 6.46 This defines the TVD region in the 0-0 plane the curve 0 0 must lie in this region, which is shown as the shaded region in Figure 6.6 a . This figure also shows the functions 0 0 Fig. 6.6. Limiter functions 0 0 . a The shaded regions shows where function values must lie for the method to be TVD. The...
b Cyr
Fig. 16.5. a Computed solution to the Buckley-Leverett problem at time t 1 using CLAWPACK with sufficiently small time steps that the Courant number is less than 1. b Entropy-violating solution obtained with CLAWPACK when the true Courant number is greater than 1. claw book chap16 bucklev In computing the fluctuations an entropy fix must be included, as in the convex case, if the solution contains a transonic rarefaction. It is also important to insure that the Courant number is less than 1....
Claw Book Chap13 Rpsoln
Fig. 13.2. Shallow water sloshing in a rectangular pan that is oscillated along the diagonal. claw book chap13 slosh background depth h0 1, then these would propagate with their shape essentially unchanged, at the characteristic speeds Vgh0 1. In Figure 13.1 the variation in depth is sufficiently large that the nonlinearity plays a clear role, and each wave shows the same behavior as the nonlinear traffic-flow example of Figure 11.1. The front of the wave relative to its direction of motion...
VariableCoefficient Linear Equations
In the preceding chapters we have primarily studied linear problems with constant coefficients. Many interesting problems involve spatially-varying coefficients. This chapter is devoted to exploring some of the issues that arise, both analytically and numerically, in this case. There are several distinct forms that a variable-coefficient hyperbolic system might take, each of which arises naturally in some contexts. One possibility is that the coefficient matrix A multiplying qx varies with x,...
Info Baj
in the quasilinear form ut uux 0 and multiply by 2u, we obtain 2uut 2u ux 0, which can be rewritten as This is again a conservation law, now for u2 rather than u itself, with flux function f u2 f u2 3 2. The differential equations 11.34 and 11.35 have precisely the same smooth solutions. However, they have different weak solutions, as we can see by considering the Riemann problem with ul gt ur. The unique weak solution of 11.34 is a shock traveling at speed S1 ul ur , 11.36 whereas the unique...
Randall J Leveque
more information - www.cambridge.org 0521810876 Finite Volume Methods for Hyperbolic Problems This book contains an introduction to hyperbolic partial differential equations and a powerful class of numerical methods for approximating their solution, including both linear problems and nonlinear conservation laws. These equations describe a wide range of wave-propagation and transport phenomena arising in nearly every scientific and engineering discipline. Several applications are described in a...
Info Lyd
3.2 Superposition of Waves and Characteristic Variables 48 3.6 Domain of Dependence and Range of Influence 50 3.7 Discontinuous Solutions 52 3.8 The Riemann Problem for a Linear System 52 3.9 The Phase Plane for Systems of Two Equations 55 3.10 Coupled Acoustics and Advection 57 3.11 Initial-Boundary-Value Problems 59 Exercises 62 4 Finite Volume Methods 64 4.1 General Formulation for Conservation Laws 64 4.2 A Numerical Flux for the Diffusion Equation 66 4.3 Necessary Components for...
Nonlinear Scalar Conservation Laws
We begin our study of nonlinear conservation laws by considering scalar conservation laws of the form When the flux function f q is linear, f q uq, then this is simply the advection equation with a very simple solution. The equation becomes much more interesting if f q is a nonlinear function of q. The solution no longer simply translates uniformly. Instead, it deforms as it evolves, and in particular shock waves can form, across which the solution is discontinuous. At such points the...
Info Oik
To connect this with the one-way wave equation 1.4 , let wl x, t p x, t pcu x, t , where c VK p. Then it is easy to check that w x, t satisfies the equation and so we see that c can be identified as the speed of sound. On the other hand, the function This is also a one-way wave equation, but with propagation speed -c. This equation has solutions of the form q2 x, t q x ct and models acoustic waves propagating to the left at the speed of sound, rather than to the right. The system 1.5 of two...
Info Btf
Fig. 11.1. Solution to the traffic flow model starting with a bulge in the density. a Trajectories of individual vehicles and the density observed by each driver. b The characteristic structure is shown along with the density as computed by clawpack. claw book chap11 congestion seen by the kth driver as qk t , as in 9.32 , then the velocity of the kth car at time t will be U qk t . This is a reasonable model of driver behavior the driver chooses her speed based only on the distance to the car...