CS205b/CME306Lecture 131 Multiple Spatial DimensionsIn multiple spatial dimensions, the ENO discretization is applied independently using a dimensionby d imen s ion 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 from 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 asimilar m an ner 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 theentire equation in time with a method of lines approach using, for example, a TVD Runge-Kuttamethod.We emphasize th at dimen sion by dimens ion discretization is not the same as dimensional split-ting, su ch 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 Fedkiw, §14.5.1In general, a hyperbolic system will simultaneously contain a mixture of processes: smooth bulkconvection and wave motion, and discontinuous processes 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 sur face, as the evolution proceeds in time these jump s willbreak up into a combination of shocks, rarefactions and contacts, in addition to any bulk motionand sound waves that may exist or develop. This is called a shock tube experiment.The hyperbolic sys tems we encounter in physical problems are written in what are effectively themixed variables wh ere the apparent behavior is quite complicated. A transformation is requiredto decouple them b ack 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 linearized over a small space and time region,and this pr ovides the basis for the theoretical understanding of the structure of general hyperbolicsystems of conservation laws. This is called a transformation to characteristic variables. As weshall see, this transformation also provides th e basis for designing app ropriate numerical methods.1uxvelocity = −1vxvelocity = 1Figure 1: The solution is the initial data for u moving to the left with speed 1, and the initial datafor v moving to the right w ith speed 1.Consider a simple hyperbolic system of N equationsφt+ [f(φ)]x= 0 (1)in one spatial dimension. The basic idea of characteristic numerical schemes is to transform thisnonlinear system to a system of N (nearly) independent scalar equations of the formφt+ λφx= 0and discretize each scalar equation independently in an upwind biased fashion using the character-istic velocity λ. T hen transform the discretized system back into the original variables.2.1 ExampleWe start with an example of two separate scalar equations and show how we can change variablesto write them as a coupled system. Consider the two equationsut− ux= 0vt+ vx= 0u(x, 0) = u0(x)v(x, 0) = v0(x)The analytic solution isu(x, t) = u0(x + t)v(x, t) = v0(x − t)For example, Figure 1 depicts the solution for the initial data given below.u0(x) =1, x ∈ (−1, 0)0, otherwisev0(x) =1, x ∈ (0, 1)0, otherwiseNext we make the change of variablesw = v + uz = v − u2wxt = 0wxt = 1Figure 2: The solution consists of two separate components, one moving to the left, and the othermoving to the right.This giveswt= vt+ ut= −vx+ ux= −zxzt= vt− ut= −vx− ux= −wxSo u and v are independent of each other, but w and z depend on each other. The system for wand z can be w ritten aswzt+zwx= 0.The solution is given byw(x, t ) = v0(x − t) + u0(x + t)z(x, t) = v0(x − t) − u0(x + t)The graph for w is sh own in Figure 2.This demonstrates that although the picture for w may appear complicated, the underlyingsolutions u and v are simply two waves m oving to the left and right.Now we rewrite the system aswzt+0 11 0wzx= 0which is in the formφt+ Jφx= 0.Similarly, we can write the system (1) in quasilinear form asφt+ f′(φ)φx= 0.Here J =∂f∂φ. Recall that in the scalar caseφt+ f (φ)x= 0where we had the quasilinear formφt+ f′(φ)φx= 03the characteristic speed was given by f′(φ). For the case of systems, the characteristic speeds aregiven by the eigenvalues of the Jacobian, J.Coming back to our example, we haveJ =0 11 0We compute the eigenvalues:det (J − λI) =−λ 11 −λ= λ2− 1So the eigenvalues of J areλ1= −1, λ2= 1.Next we determine the eigenvectors. For λ1= −1, we have JR1= λ1R1. We solve for R1=ab.0 11 0ab= −ab⇒ba= −abHence R1=1−1is a solution. For λ2= 1, we have JR2= λ2R2. We solve for R2=cd.0 11 0cd=cd⇒dc=cdHence R2=11is a solution. Therefore, we have computed thatJR1R2=R1R2λ100 λ2.Note th at if R =R1R2is a matrix whose columns are the right eigenvectors of J, andL = R−1is taken to be the m atrix whose rows are left eigenvectors of J,JR = RΛ LJ = ΛL LR = RL = I.In summary, we have computed the eigensystem for our example, and we can use this totransform J into diagonal form,LJR = Λ.It is important to note that if a system is hyperbolic, J will have N real eigenvalues λp, p =1, . . . , N, and N linearly independent right eigenvectors. Once the eigensystem is determined, wecan use it to diagonalize the matrix J.Suppose we want to discretize our equation at the node x0, where L and R have values L0andR0. To get a locally diagonalized form, we multiply our system equation by the constant matrixL0which n early diagonalizes J over the region near x0. We require a constant matrix so that wecan move it inside all derivatives to obtain[L0φ]t+ L0JR0[L0φ]x= 04where we have inserted I = R0L0to put the equation in a more recognizable form. The spatiallyvarying matrix L0JR0is exactly diagonalized at the point x0, with eigenvalues λ0,p, and it is nearlydiagonalized at nearby points. Thus the equations are sufficiently decoupled for us to apply upwindbiased discretizations