# 1 PART1: Bratu problem

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1 D9: Advanced Numerical Analysis: 3 Computer Assignment 3: Recursive Projection Method(RPM) Joakim Möller PART: Bratu problem. Background The Bratu problem in D is given by xx u + yy u + λe u =, u Γ = () By use of finite differences the discrete solution can be obtained from the fixed point iteration Au n+ i j = f (u n i j), A = [D +x D x + D +y D y ], f (u n i j) = λe un i j, () where D + and D are the forward and backward operators in the x- and y-directions. By using that A is a penta-diagonal symmetric positive definite, the solution u n+ i j can easily be obtain via the Cholesky factorization, A = C T C. This only N p +3p flops and N square roots, see []. To simplify the notations we write the fixed-point iteration as u n+ = F(u n ) (3) The ultimate convergence rate depends on the spectral radius of the Jacobian J(u ) F(u ), u (4) ρ(j) max µ j j where µ j, j =,...,N are the eigenvalues of J, and u is the final solution, u = F(u ) The spectral radius is closely related to the value of the parameter λ, see below

2 u(λ) λ (a) ρ(λ) λ (b) Figure : The spectral radius and solution norm (Euclidean) as a function of the parameter λ. The graphs was obtain by using Pseudo-arclength RPM-continuation, see [6].

3 We see from the solution branch that for each λ-value, there is both a stable and unstable solution. The fixed point iteration can only compute the solutions on the lower part of the branch. Furthermore the convergence rate will approach zero as λ approach 6.88, the turning point where ρ(j) becomes one. In order to increase the convergence rate as λ approaches the critical value and even compute unstable solutions we will use the Recursive Projection Method (RPM), [4], [5], [6]. q n+ = ( I V p Vp T ) F(p n + q n ) p n+ = p n ( ) V p I V T V T p JV p p (F(u n ) p n ) (5) u n+ = q n + p n where p n and q n is the solution projected on the dominant eigenspace, spanned by V p, associated with the p largest eigenvalues, and its orthogonal complement. The lab consists of parts: Part, we will investigate a linear PDE to examine some theoretical properties of RPM. Part, we compute the solution of () for three different λ-values. For two λ-values the solution will be stable, whereas in the last case, ρ(j) > and special care has to be taken. The stable cases will be started with a random initial vector. whereas the last case the initial solution is taken from the upper-part of the branch. Run the Matlab-function u=initfield(x,y ), where X and Y are two matrices containing the grid-points of your mesh. This function will create the initial data for the internal points of the computational by interpolation of the data contained in the file init.mat... Linear test problem Implement the RPM-algorithm. The basis extraction is done via the QR-method. Use a Krylov space with vectors. Test RPM with the following linear problem. Use an equal meshing in x and y, where λ = 9.6. N x = N y = 3, xx u + yy u + λ( + u) = f (x,y), f (x,y) = sin(πx)y( y) (6) Plot the residuals of (3) and p n,q n and of (5) 3

4 Use Arnoldi to compute the largest eigenvalues. Reproduce the the eigenvalue plot below for the largest eigenvalues, mark which eigenvalues that RPM extracted. How do the eigenvalues relate to the convergence plot. largest eigenvalues for lambda= Figure : The largest eigenvalues of (3) associated with (6). Plot u(x, y), p(x, y), q(x, y), when the solution has converged.. Non-linear test problem Run the three test cases given in the table below: CASE λ u initial solution stable rand yes rand yes file no Table : Nonlinear test-cases In order to obtain convergence for the third case, the unstable eigenvalue must first be eliminated. Use Arnoldi to compute the unstable eigenvalue and eigenvector. Use this vector to form the first component of the basis V p, i.e. we will apply RPM directly from the first iteration, and not wait for the QR-method to identify the dominant eigenspace. Plot residuals, spectra of the largest eigenvalues and u(x, y), p(x, y), q(x, y) when the converge solution. 4

5 PART : Compressible Flow Past a Symmetric Airfoil.. Background The governing equations of fluid dynamics are known as the Navier-Stokes equations. t (W ) + x ( f f v) + y (g g v) + z (h h v) = (7) where t denotes the time. The state-vector W is given by W = and the convective fluxes are defined as ρ ρu ρv ρw ρe (8) f = ρu ρu + p ρuv ρuw u(ρe + p),g = ρv ρvu ρv + p ρvw v(ρe + p),h = ρw ρwu ρwv ρw + p w(ρe + p) (9) Here ρ is the density, u, v and w are the Cartesian velocity components, p is the pressure and E is the total energy. The viscous fluxes are defined as f v = τ xx τ xy τ xz (τu) x q x,g v = with the shear stress tensor τ given by τ yx τ yy τ yz (τu) y q y,h v = τ zx τ zy τ zz (τu) z q z () 5

6 ) τ xx = 3 ( µ u x v y w z τ yy = 3 ( µ u x + v y w z τ zz = 3 ( µ u x v y + w z ) τ xy = τ yx = µ( v x + u y τ xz = τ zx = µ( w x + u z τ yz = τ zy = µ( v z + w y ) ) ) ) () where µ is the viscosity (Stokes hypothesis). The viscous dissipation in the energy equation is calculated from (τu) x = τ xx u + τ xy v + τ xz w (τu) y = τ yx u + τ yy v + τ yz w (τu) z = τ zx u + τ zy v + τ zz w () and the heat flux due to conduction is calculated according to Fourier s law, q x = k T x q y = k T y q z = k T z (3) where T is the temperature and k the heat conductivity. Assuming a constant Prandtl number (for air Pr =.7), the heat conductivity can then be found by k = µc p /Pr The specific heats at constant volume and constant pressure are constant for a caloric perfect gas, and can be calculated from c v = R/(γ ) and c p = γc v respectively, with γ =.4, and R the gas constant, equal to 87 (J/kgK) for air. To close the system of equations the pressure p must be related to the state vector W. This relation depends on the model used to describe the thermodynamic properties of the gas. For a caloric perfect gas this relation states p = ρe(γ ) = ρc v T(γ ) = ρrt (4) 6

7 Free stream mach number: M =.65. Free stream mach number: M = Mach Number Mach Number where e is the internal energy. The internal and total energy are related by e = E ( u + v + w ) (5) The equations are discretized using the MacCormack scheme, second order accurate. Here we will restrict the simulation to a stationary inviscid D flow, that is the viscosity is set to zero. The test case is the the transonic- and supersonic flow over a symmetric airfoil. 7 x grid y x Figure 3: The Euler mesh. 7

8 . Setup The students will be given a compressible Euler code implemented in matlab. The task is to connect the Euler code as a black-box to the RPM code, which will be used to accelerate the convergence. In doing so some alteration of the code must be performed. rewrite the file NS.m as a function-file of the form w n+ = F(w,mode,nstep) (6) where w = w(ρ,ρu,ρw,ρe) T for the mass, momentum and energy fluxes. mode is a flag giving information how the function file should operate. If mode =, do initialization. That is the code should create an initial solution and set other parameters. In order not to reset all data in every time step, use matlabs save and load of the workspace. Be sure that w is not contained in your saved work space, since this variable is to be sent into the code. If mode = compute the solution. The parameter nstep indicates the number of time steps to be taken with in the code before returning the state vector to RPM. RPM works on a vector, whereas the code NS-code work with matrices, hence use the matlab function reshape to alter the structure of w each time it is sent in and out of the code function w=f(w,mode,nstep) if mode== initialize w save data reshape w as a vector elseif mode== load data reshape w as a matrix for i=:nstep end Maccormack scheme save data 8

9 end reshape w as a vector The state vector is usually very scew, ρe is usually several order of magnitudes larger than the other components of the state vector. In order to improve the performance of RPM we construct a function that scales the state vector. w=scale(w,flag), where if flag= the vector is scaled and if flag=, the vector is unscaled. Note that the vector should only be scaled within RPM whereas when sent to F it should be unscaled, hence the structure of the function used by RPM, should be of the form function w=func(w,state,nstep) w=scale(w,) w=f(w,state,nstep) w=scale(w,) Note that the scale function also must be added in the beginning and at the end of the RPM algorithm. The scaling is done as follows: ρ = ρ ρ ρu = ρv = ρu ρ p ρv ρ p (7) ρe = ρe γ+ (γ ) p where ρ and p are the initial density and pressure, and γ =.4 is the ratio between the specific heats for air..3 The lab Exercise Set the free stream mach-number to.65 and run the code. Set the amount of artificial viscosity in NS to EPS = [.5,.5]; EPS4 = [.8,.8] and a CFL number equal.8. A good nstep value is 5. Typical values for the Krylov acceptance ratio k a is 3 and the size of the Krylov matrix is. How does these affect the convergence rate and basis size? 9

10 Plot the convergence history. How large was the extracted basis? Plot also the convergence history, that is the projected residuals and the total residual Pr(u n ), Qr(u n ), r(u n ) and compare this with the convergence history for the original problem, that is if RPM is not switched on, see below Supersonic wing, M=.65 Original r(u) Pr(u) Qr(u) 4 residual iter Figure 4: The convergence history for case M =.65. Plot solutions, for example the density and its projections, Pρ and Qρ. Use eigs (or your own Arnoldi implementation ) to plot the largest eigenvalues. Within this plot, plot the eigenvalues of PJP, found by RPM. Also plot maybe the to or 3 first eigenvectors of J. Can you see any resemblance between these and the solution. To plot the eigenvectors of J, use that JVy = V Hy + Ry (8) where y is an eigenvector of H and hence the vector x = Vy is an approximative eigenvector of J. Note that the eigenvectors might be complex, so given to complex conjugates eigenvectors, the change them to [v, v] [R(v), I(v)]

11 Redo the simulations with some different values for k a and k s, does it effect the basis. Redo the simulation with M =.7. Use EPS = [.5,.5] and EPS4 = [.,.] and a CFL number equal.8. 3 PART 3 Assignments for experts: Arclength continuation with RPM Heat and mass transfer in porous spherical catalyst where a first-order reaction occurs is described by a non-linear boundary value problem, d y dξ + a ( ) dy µβ( y) ξ dξ = φ yexp, ξ (9) + β( y) subject to boundary conditions ξ =, dy dξ () = ξ =,y() = For technical purposes usually a plot in η φ is drawn, where η, the effectiveness factor, is defined as η = a + φ dy (ξ = ) dξ Assignment: Using RPM, compute the bifurcation diagram, with the parameter settings a =, µ = 6, β =.5. φ [,.6). In order to pass the turning point, i.e. when max(λ) =, RPM has to be modified. Use the Pseudo-arclength RPM, to handle this task, see [6]. In order for the continuation to work, it might be necessary to update the extracted basis before the computation of the solution at a new φ-value. Use e.g. Arnoldi with DGKS for this purposes, see [4] In Fig(5), the bifurcation diagram for the parameter setting is shown. We used a second order finite difference scheme explicit in the forcing term and implicit in the diffusion term. Using 3 grid points, the equation had different solutions. As the number of grid points increased, so did the number of solutions Fig(5), here we used 3 grid-points, one can at most see 8 solutions around φ =.. In [], one claims that the equation had 5 solutions for this parameter setting. Due to the lack of memory, one can not continue to refine the grid using second order

12 methods. Therefore, you should implement a suitable higher order method, to see if you can reproduce the results of []. If you solve this problem, you do not need to solve the other computer assignments. Good look

13 4.5 η(φ) (a) η φ-plot max λ(φ) (b) Max. eigenvalue of the Jacobian Figure 5: The bifurcation diagram of (9), non-resolved.

14 References [] Childs B., Scott M., Daniel W., Denman E., Nelson P., Codes for Boundary- Value Problems in Ordinary Differential Equations, Lecture Notes in Computer Science 76, Springer-Verlag, 97. Lecture Notes in Computer S [] Golub G., Van Loan C., Matrix Computations, The Johns Hopkins University Press, Baltimore, third edition, 996 [3] Hanke M., Lecture Notes: Advanced Numerical Methods, Royal Institute of Technology, NADA, February [4] Möller J., Lecture Notes: Eigenvalue problems, and Applications, Royal Institute of Technology, NADA, to appear [5] Möller J., Studies of Recursive projection Methods for Convergence acceleration of Steady State Calculations. TRITA-NA-9, Royal Institute of Technology, Stockholm, Sweden, June [6] Shroff G. M., Keller B., Stabilization of Unstable Procedures: The Recursive Projection Method, SIAM J. Numer. Anal. vol 3, No 4,pp. 99-, August 993 4

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