CS205b/CME306Lecture 120.1 ENO-Roe DiscretizationFor a specific cell wall, located at xi0+1/2, we find the associated numerical flux function Fi0+1/2as follows. First, we define a characteristic speedλi0+1/2= f′(φi0+1/2).For example, recall Burgers’ equation,ut+u22x= 0.The flu x is given byf(u) =u22f′(u) = u.Therefore,λ(x) = f′(u(x)) = u(x).The value of u at the half grid points is defi ned using a standard linear averageui0+1/2= (ui0+ ui0+1)/2.Then, if λi0+1/2> 0, set k = i0. Otherwise, set k = i0+ 1. Next, defineQ1(x) = (D1kH)(x − xi0+1/2).If |D2k−1/2H| ≤ |D2k+1/2H|, then c = D2k−1/2H and k⋆= k − 1. Otherwise, c = D2k+1/2H andk⋆= k. DefineQ2(x) = c(x − xk−1/2)(x − xk+1/2).If |D3k⋆H| ≤ |D3k⋆+1H|, then c⋆= D3k⋆H. Otherwise, c⋆= D3k⋆+1H. DefineQ3(x) = c⋆(x − xk⋆−1/2)(x − xk⋆+1/2)(x − xk⋆+3/2).ThenFi0+1/2= H′(xi0+1/2) = Q′1(xi0+1/2) + Q′2(xi0+1/2) + Q′3(xi0+1/2)which simplifies toFi0+1/2= D1kH + c(2(i0− k) + 1)∆x + c⋆(3(i0− k⋆)2− 1)∆x2.10.2 ENO-LLF Discretization (and the Entropy Fix)The ENO-Roe discretization can admit entropy violating expansion shocks near sonic points. Thatis, at a place where a characteristic velocity ch anges sign (a sonic point) it is possible to have astationary expansion shock solution with a discontinuous jump in value. If this jump were smoothedout even slightly, it would break up into an expansion fan (i.e. rarefaction) and dissipate, whichis the desired phy s ical solution. For a specific cell wall, xi0+1/2, if there are no nearby sonicpoints, then we use the ENO-Roe discretization. Otherwise, we add high order dissipation to ourcalculation of Fi0+1/2to break up any entropy violating expansion shocks. We call this entropyfixed version of the ENO-Roe discretization ENO-Roe Fix or just ENO-RF. More specifically, weuse λi0= f′(φi0) and λi0+1= f′(φi0+1) to decide if th ere are sonic points in the vicinity. If λi0and λi0+1agree in sign, we use th e ENO-Roe discretization where λi0+1/2is taken to be the samesign as λi0and λi0+1. Otherwise we u s e the ENO-LLF entropy fix discretization given below. Notethat ENO-LLF is applied at both expansions where λi0< 0 and λi0+1> 0 and at shocks whereλi0> 0 and λi0+1< 0. While this add s extra numerical dissipation at shocks, it is not harmf ulas shocks are self-sharpening. In fact, this extra dissipation provides some viscous regularizationwhich is especially desirable in multiple spatial dimensions. For this reason, authors sometimesuse th e ENO-LLF method everywhere as opposed to mixing in ENO-Roe discretizations where theupwind direction is well determined by the eigenvalues λ.The ENO-LL F discretization is formulated as follows. Consider two primitive functions H+and H−. We compute a divided difference table for each of them with their firs t divided differencesbeingD1iH±= f(φi) ± αi0+1/2φiαi0+1/2= max(|λi0|, |λi0+1|)is our dissipation coefficient, and controls th e amount of dissipation added. Note that the dissi-pation coefficient, αi0+1/2, is determined locally for each cell wall, h en ce the name ENO-lo cal LaxFriedrichs. (One could also construct a scheme where a global dissipation coefficient α is used, butthis generally adds too much dissipation).The second and third divided differences, D2i+1/2H±and D3iH±are then defined in the standardway, like those of H.For H+, set k = i0. Then, replacing H with H+everyw here, define Q1(x), Q2(x), Q3(x), andfinally F+i0+1/2using the ENO-Roe algorithm above. For H−, set k = i0+ 1. Th en , replacing Hwith H−everyw here, define Q1(x), Q2(x), Q3(x), and finally F−i0+1/2again by using the ENO-Roealgorithm above. Finally,Fi0+1/2=F+i0+1/2+ F−i0+1/22is the new numerical flux function with added high order d issipation.1 Multiple Spatial DimensionsIn multiple spatial dimensions, the ENO discretization is applied independently using a dimensionby dimension discretization. For example, consider a two dimensional conservation lawφt+ f(φ)x+ g(φ)y= 0on a rectangular 2-D grid. Here, we sweep through the grid f rom bottom to top performing ENOon 1-D horizontal rows of grid points to evaluate the f(φ)xterm. The g(φ)yterm is evaluated in a2similar manner sweeping through the grid from left to right performing ENO on 1-D vertical rowsof grid points. Once we have a numerical approximation to each of the spatial terms, we update th eentire equation in time w ith a method of lines approach using, f or example, a TVD Runge-Kuttamethod.We emphasize that dimension by dimension discretization is not the same as dimensional s plit-ting, such as the first order Godunov splitting and second order Strang splitting. In dimension bydimension discretization, the fluxes in each dimension are evaluated independently, but the timestepping is still coupled.2 Systems of Conservation LawsSupplementary Reading: Osher and Fed kiw , §14.5.1In general, a hyperbolic system will simultaneously contain a mixture of processes: smooth bulkconvection and wave m otion, and discontinuous pr ocesses involving contacts, shocks and rarefac-tions. For example, if a gas in a tube is initially prepared with a jump in the states (density,velocity and temperature) across some surface, as th e evolution proceeds in time these jumps willbreak up into a combination of shocks, rarefactions and contacts, in addition to any bulk motionand sound waves that may exist or d evelop.The hyperbolic systems we encounter in physical problems are written in wh at are effectively themixed variables where the apparent behavior is quite complicated. A transformation is requiredto decouple them back into unmixed fields that exhibit the pure contact, shock and rarefactionphenomena (as well as bulk convection and waves). In a real system, this perfect decoupling is notpossible because the mixing is nonlinear, but it can be achieved approximately over a small spaceand time region, and this provides the basis for the theoretical understanding of the structure ofgeneral hyperbolic systems of conservation law s. Th is is called a transformation to characteristicvariables. As we shall see, this transformation also provides the basis for designing appropriatenumerical methods.Consider a simple hyperbolic system of N equationsφt+ [f(φ)]x= 0 (1)in one spatial dimension. The basic idea of ch aracteristic numerical schemes is to transform thisnonlinear system to a system of N (nearly) independent scalar equations of the