Lecture 3Wednesday, April 6, 2005Supplementary Reading: Osher and Fedkiw, Section 14.1Previously we motivated our study of numerical methods for hyperbolic con-servation laws by focusing on the linear advection equation, φt+~V · ∇φ = 0.We looked at various methods, including 1st, 2nd, and 3rd order HJ ENOfor the spatial derivative and 1st, 2nd, and 3rd order TVD RK for the tem-poral derivative. We now shift our focus to solving hyperbolic conservationlaws.1 Hyperbolic Conservation LawsA continuum physical system is described by the laws of conservation ofmass, momentum and e nergy. The integral form of the conservation law isderived by considered a fixed control volume (or re gion in two dimensions).Let us denote the control volume by Ω, and its boundary by ∂Ω. if urepresents the conserved quantity, then the total amount of u in the controlvolume is given byZΩu dVThe rate of change of the total amount in the control volume is given by theflux through the region boundary, plus whatever internal sources exist.ddtZΩu dV = −Z∂Ω~f(u) · dA +ZΩs(u) dVThe flux can be e ither convective or diffusive. The distinction is that dif-fusive fluxes are driven by gradients, while convective fluxes persist even inthe absence of gradients. As an example of a diffusive flux, consider theopening of a perfume bottle. The gradient in concentration of the perfumecauses it to diffuse. For most flows where compressibility is important, e.g.1flows with shock waves, one only needs to model the convective transportand can ignore diffusion (mass diffusion, viscosity and thermal conductivity)as well as the source terms (such as chemical reactions, atom ic excitations,and ionization processes). Moreover, convective transport requires special-ized numerical treatment while diffusive and reactive effects can be treatedwith standard numerical methods, such as simple central differencing, thatare independent of those for the convective terms. A source term mightinclude creation of the quantity through a chemical reac tion. Conservationlaws with only convective fluxes are known as hyperbolic conservation laws.The weak form of the conservation law is usually written asddtZΩu dV +Z∂Ω~f(u) · dA =ZΩs(u) dVThe equation now resembles the linear advection equation we looked atpreviously.We now consider the strong form, or differential form of the conservationlaw. The strong form can be derived from the weak form by taking aninfinitesimally small control volume and applying the divergence theorem.The equation is then written as∂u∂t+ ∇ ·~f(u) = s(u)The strong form may not always hold, as it requires that∂u∂tand ∇·~f(u) exist.The strong form is not valid when there is a s hock, contact discontinuity,or when the function is not smooth. These are the types of phenomena wewould like to consider.The first thing to realize is that the presence of discontinuities poses alimitation on the order of accuracy of any numerical scheme we might devise.There is a conjecture that states that we cannot get a scheme with higherthan first order accuracy. However, we are still interested in higher ordermethods such as ENO, because although our scheme is limited to first orderaccuracy overall, in many parts of the domain the dominant error term willbe the higher order one. For example, we may have an error that looks likeC14x + C24x3, with C1 C2almost everywhere in our domain. This iscalled a high resolution method.The important physical phenomena exhibited by hyperbolic conservationlaws are1. bulk convection and waves2. contact discontinuities23. shocks4. rarefactions.We briefly describe the physical features and mathematical model equa-tions for each effect, and most importantly note the implications they haveon the design of numerical methods.1.1 Bulk Convection and WavesBulk convection is simply the bulk movement of matter, carrying it fromone spot to another, like water streaming from a hose. Waves are smallamplitude s mooth disturbances that transmit through the system withoutany bulk transport like ripples on a water surface or sound waves through air.Whereas convective transport occurs at the gross velocity of the material,waves propagate at the “speed of sound” in the system (relative to the bulkconvective motion of the system). Waves interact by superp os ition, so thatthey can either cancel out (interfere) or enhance each other.The simplest model equation that describes bulk convective transport isthe linear convection equationut+ ~v · ∇u = 0 (1)where ~v is a constant equal to the convection velocity. The solution to thisis simply that u translates at the constant speed ~v. This same equation canalso be taken as a simple model of wave motion, if u is a sine wave and ~v isinterpreted as the speed of sound. The linear convection equation is also animportant model for understanding smooth transport in any conservationlaw. As long as~f is s mooth and u has no jumps in it, the general scalarconservation lawut+ ∇ ·~f(u) = 0 (2)can be rewritten asut+~f0(u) · ∇u = 0 (3)where~f0(u) acts as a convective velocity. That is, locally in smooth parts ofthe flow, a conservation law behaves like bulk convection with velocity~f0(u).This is called the local characteristic velocity of the flow. For systems, theterm~f0(u) is the Jacobian∂~f∂~u.3Note: One must be careful in going from equation from (2) to (3). Indoing so, we are assuming that f depends on x through u only. For example,consider the equation for c onservation of mass in one dimension.ρt+ (ρu)x= 0The chain rule givesρt+ uρx+ ρux= 0However, applying (3) with f(ρ) = ρu would giveρt+ uρx= 0which is a linearization. It assumes that ux= 0, or that f depends on xthrough ρ only. The linearized equation is incorrect. However, it can beused as an aid in developing intuition and as a guide for devising numericalschemes.1.2 Contact DiscontinuitiesA contact discontinuity is a persistent, discontinuous jump in mass densitymoving by bulk convection through the system. Since there is negligiblemass diffusion, such a jump persists. These jumps usually appear at thepoint of contact of different materials, for example, a contact discontinuityseparates oil from water. Contacts move at the local bulk c onvection speed,or more generally the characteristic speed, and can be modeled by using step-function initial data in the bulk convection equation 1. Since contacts aresimply a bulk convection effect, they retain any perturbations they