CS205b/CME306Lecture 101 Hyperbolic Conservation LawsSupplementary Reading: Osher and Fedkiw, Section §14.1.1, §14.1.2, §14.1.3, §14.1.4; Leveque§11.6, §12.9, §12.10, §12.11A physical system is described by the laws of conservation of mass, momentum and energy. Theintegral form of the conservation law is derived by considering a fixed control volume. Let us denotethe control volume by Ω , an d its boundary by ∂Ω. If φ represents the conserved quantity, then thetotal amount of φ in the control volume is given byZΩφ dVThe rate of change of φ in the control volume is given by th e flux through the region boundary,plus whatever internal sources exist.ddtZΩφ dV = −Z∂Ωf(φ) · dS +ZΩs(φ) dVThe flux can be either convective or diffusive. The d istinction is that diffusive fluxes are drivenby gradients, while convective fluxes persist even in the absence of gradients. As an exampleof a diffusive flux, consider the opening of a perfume bottle. The gradient in concentration ofthe perfume causes it to diffuse. For most flows where compressibility is important, e.g. flowswith shock waves, one needs only to model the convective transport and can ignore diffusion(mass diffusion, viscosity and thermal conductivity) as well as the s ource terms (such as chemicalreactions, atomic excitations, and ionization processes). Moreover, convective transport requiresspecialized numerical treatment while diffusive and reactive effects can be treated with standardnumerical methods—such as simple central differencing—that are independent of th ose for theconvective terms. A source term might includ e creation of the quantity thr ough a chemical r eaction.Conservation laws with only convective fluxes are known as hyperbolic conservation laws.The weak form of the conservation law is usually written asddtZΩφ dV +Z∂Ωf(φ) · dS =ZΩs(φ) dVThe equation now resembles the linear advection equation we looked at previously.We now consider the strong form (or differential form ) of the conservation law. The strong formcan be derived from the weak form by taking an infinitesimally small control volume and applyingthe divergence theorem. The equation is then written asφt+ ∇ · f (φ) = s(φ)1The str ong form may not always hold, as it requires that ∇ · f (φ) exist. The strong form is notvalid when there is a shock, contact discontinuity, or when the function is not smooth. These arethe types of phenomena we would like to consider.The first thing to realize is that the presence of discontinuities poses a limitation on the orderof accuracy of any numerical scheme we might devise. There is a conjecture that states that wecannot get a scheme with higher than fir s t order accuracy. However, we are still interested in higherorder methods such as ENO, because even if our scheme is limited to first order accuracy overall,in many parts of the domain the dominant error term will be the higher order one. For example,we may have an error that looks like C1∆x + C2∆x3, with C1≪ C2almost everywhere in ourdomain, so th at the higher order term dominates for the time step size taken. This is called a highresolution method.The important physical phenomena exhibited by hyperbolic conservation laws are:1. bulk convection and waves2. contact discontinuities3. shocks4. rarefactionsWe will briefly describe the physical features and mathematical model equations for each effect,and most importantly note the implications they have on the design of numerical methods. Thefirst two phenomena are linearly degenerate because they can be modeled locally by the linearadvection equation. The last two phenomena are genuinely nonlinear.1.1 Bulk Convection and WavesBulk convection is simp ly the bulk movement of matter from one spot to another, like waterstreaming from a hose. Waves are small amplitude smooth disturbances that transmit through thesystem without any bulk transport, like ripples on a water surface or sound waves through air.Whereas convective transport occurs at th e gross velocity of the material, waves propagate at thespeed of sound in the system (relative to the bulk convective motion of the system). Waves interactby superposition, so that they can either cancel out (interfere) or enhance each other.The simplest model equation that describes bulk convective transport is the linear convectionequationφt+ u · ∇φ = 0 (1)where u is a constant equal to the convection velocity. The solution to this is simply that φtranslates at the constant speed u. This same equation can also be taken as a simple model ofwave motion, if φ is a sine wave and u is interpreted as the speed of s ou nd. The linear convectionequation is also an important model for understanding smooth transport in any conservation law.As long as f is smooth and φ has no jumps in it, th e general scalar conservation lawφt+ ∇ · f (φ) = 0 (2)can be rewritten asφt+ f′(φ) · ∇φ = 0 (3)2where f′(φ) acts as a convective velocity. That is, locally in smooth parts of the flow, a conservationlaw behaves like bulk convection with velocity f′(φ). This is called the local characteristic velocityof the flow. For systems, the term f′(φ) is the Jacobian∂f∂u.Note, however, that one must be extremely careful going from equation 2 to equation 3. Indoing so, we are assuming that f depends on x through φ only. For example, consider the equationfor conservation of mass in on e dimension.ρt+ (ρu)x= 0.The chain rule givesρt+ uρx+ ρux= 0.However, applying (3) with f(ρ) = ρu would giveρt+ uρx= 0which is a linearization. It assumes that ux= 0, or that f depends on x through ρ only, which willnot be true in general. However, this formulation can be used to gain intuition and as a guide fordevising numerical schemes.1.2 Contact DiscontinuitiesA contact discontinuity is a persistent, discontinuous jump in mass density moving by bulk convec-tion thr ough th e system. Since there is negligible mass diffusion, such a jump persists. These jumpsusually appear at the point of contact of different materials, for example, a contact discontinuityseparates oil fr om water. Contacts move at the local bulk convection speed, or more generally thech aracteristic speed, and can be modeled by using step-function initial data in equ ation 1. Sincecontacts are simply a bulk convection effect, they retain any perturbations they receive. Thus weexpect contacts to be especially sensitive to numerical methods, i.e. any spurious