Contents of lecture 2b. Lectures 2a & 2b. Physical vs. computational coordinates [2] Physical vs. computational coordinates [1]

Transcription

1 Contents of lecture b Lectures a & b P. A. Zegeling Mathematical Institute Utrecht University The Netherlands Parameter-free non-singular moving grids in D: Theory & properties Application to resistive Magneto-hydrodynamics Application to the D Euler equations March 7, 9 Extension to 3D and application to blow-up models Physical vs. computational coordinates [] Physical vs. computational coordinates [] The D adaptive grid can also be seen as an approximation of a coordinate transformation between computational coordinates U steep solution V mild solution (ξ, η) T Ω c := [, ] [, ] Φ (with a uniform grid partitioning) and physical coordinates (x, y) T Ω p R Y η (with a non-uniform adaptive grid) non uniform mesh X uniform mesh ξ

2 Winslow s method The adaptive grid seen as a system of springs In a variational setting, a grid-energy functional (à la Winslow) can be defined as E = ( ) T x ω x + T y ω y dξdη, Ω c X(i,j) ω (i,j) where = ( ξ, η )T and ω > is a monitor function. Minimizing the energy E yields the Euler-Lagrange equations: F (ω x) = (ω y) = F F on Ω c = [, ] [, ] with BCs: x ξ= = x L, y η= = y L, x ξ= = x R, y η= = y U, x n ξ= = x n ξ= = y n η= = y n η= =. F Regularity of the transformation in D A few additional properties of the D grid Theorem [Clément, Hagmeijer & Sweers, 96]: Let ω c >, ω C, (Ω c ) and ω ξ, ω η C γ ( Ω c ), for some γ (, ). unique solution (x, y) C ( Ω c ), which is a bijection from Ω c into itself. Moreover, the acobian satisfies: = x ξ y η x η y ξ >. The three main ingredients of proof are: Carleman-Hartman-Winter Theorem (3D??) ordan Curve Theorem (3D??) Maximum principle for elliptic PDEs (3D ok!) Equidistribution in D: x ξ }{{} ω = cst Winslow in D: (x ξ ) (y η ) (x η ) (y ξ ) = ω = cst (remember: = x ξ y η x η y ξ ) the transformation behind Winslow s method is not a harmonic mapping, but it is related to it a counterexample can be given for the 3D (harmonic) case, for which the transformation looses its regularity (Liao et al 94) several components in the proof of Theorem 3 can not be applied in 3D either...

3 The monitor function ω Smoothness of the new monitor function Arclength-type (AL-) monitor: ω = + α u u α is a (problem-dependent) adaptivity -parameter which controls the amount of adaptivity. New monitor (Beckett & Mackenzie ): ω = α(t) + u m, with α(t) = u m dξdη Ω c m = : better scaling and more adaptivity than for m = Define (Huang ) ω = + γ u ( γ) RR Ωc u dξdη Ω γ = c ω dξdη Ω c ω dξdη with γ [, ) For γ =, we have ω = + RR u Ωc u 5% of the grid dξdη points is concentrated in regions of high spatial derivatives, since Ω c ω dξdη the total # of grid points Ω c ω dξdη the # of grid points in the steep layer = a smoother distributed grid than for the AL-monitor (with constant α) and α(t) is automatically computed! Smoothing of the AL-monitor function Resistive Magneto-HydroDynamics With the new monitor, application of a filter or smoother to the grid or monitor values is not necessary. Normally, smoother transitions in a general non-uniform grid can be obtained (and are needed!) by working with the smoothed value S(ω i+,j+ ) = 4 ω i+,j+ + 8 (ω i+ 3,j+ + ω i,j+ + ω i+,j+ 3 +ω i+,j ) + 6 (ω i,j + ω i,j+ 3 + ω i+ 3,j + ω i+ 3,j+ 3 ) In the numerical experiments we denote this with filter on or filter off (working merely with ω i+,j+ values i.e. S(ω) = ω). (ρv) ρ + (ρv) = + (ρvv BB) + p tot = e + (ve + vp tot BB v) = η m ( B) B + (vb Bv) = η m B p tot = p + B, p = (γ )(e ρv B )

4 Applications of models Kinematic flux expulsion in D [] B 8πµ ρ v B = (v B) + η m B with v = z =, B = (B, B, ), B = B B B = η m B + v y v B y + B v y B v y B = η m B v x + v B x B v x + B v x Kinematic flux expulsion in D [] Kinematic flux expulsion in D [3] B := A, where A := vector potential A = v ( A) η m ( A) B 3 = ( A = A ) : Four-cell convection (Weiss 66): v (x, y) = sin(πx) cos(πy) v (x, y) = cos(πx) sin(πy) A 3 A 3 = v x v A 3 y + η m A 3 < η m, A 3 t= = x where v (x, y) and v (x, y) are given and satisfy A 3 x= =, A 3 x= =, A 3 y= = A 3 y= v x + v y =

5 Global solution behaviour [] Global solution behaviour [] < η m write formal asymptotic expansion in η m : A 3 (x, y, t) = a 3,j (x, y, t) ηm j j= Substitute expansion into PDE and check first-order term (setting η m = ); this gives hyperbolic PDE: a 3, = v a 3, initial solution stays constant on sub-characteristics given by: As v =, the only critical points in ODE system are center points or saddle points ( field amplification ). At some point of time, the solution a 3, (x, y, t) can not satisfy the boundary conditions of the original PDE model: boundary (and internal) layers are formed of width O( η m ) ( magnetic flux concentrates at edges of convective cells ). For t the solution reaches a non-trivial steady-state, in which the diffusion and convection terms settle down to an equilibrium. (ẋ, ẏ) T = v Global solution behaviour [3] The algorithm for the D model S S S C C advection-diffusion-reaction PDE decoupling of physical and grid PDEs grid PDEs: system of heat eq s with artificial time S S S fixed time steps t C C Implicit-explicit time integration: -SBDF S S S freezing of non-linear terms in PDEs BiCGstab + diagonal preconditioning

6 Transformation of the PDE model [] Transformation of the PDE model [] A 3 A 3 = v x v A 3 y + η m A 3 ξ = ξ(x, y, t), η = η(x, y, t), θ = t Using the chain rule of differentiation: We can also find that A 3,t = A 3,θ θ t + A 3,ξ ξ t + A 3,η η t. Using a similar relation for η t gives us A 3,t = A 3,θ + A 3,ξ (x ηy θ x θ y η ) + A 3,η (x θy ξ x ξ y θ ). Since ξ x = y η, ξ y = x η, η y = x ξ and η x = y ξ, we find for the first-order spatial derivative terms: A 3,x = (A 3,ξy η A 3,η y ξ ) Recall: ξ t = x θ ξ x y θ ξ y = (x θy η y θ x η ) and A 3,y = (A 3,ηx ξ A 3,ξ x η ). = x ξ y η x η y ξ Transformation of the PDE model [3] Numerical results; four-cell convection [] PDE in computational coordinates becomes: Weiss model, Four cell convection; velocity field (u(x,y),v(x,y)) T 3 A 3,θ + A 3,ξ (x η y θ x θ y η ) + A 3,η (x θ y ξ x ξ y θ ) 5 = A 3,ξ ( v y η + v x η ) + A 3,η (v y ξ v x ξ )+ η m [( x η+y η A 3,ξ) ξ ( x ξx η +y ξ y η A 3,η ) ξ Y β(t) 5 ( x ξx η +y ξ y η A 3,ξ ) η + ( x ξ +y ξ A 3,η) η ] = x ξ y η x η y ξ X # of TIMESTEPS

7 D case D Euler Numerical results; four-cell convection [] D case D Euler y x y x y D case x D Euler The D Euler equations ρ ρu + ρv x E ρu ρu + p + y ρuv u(e + p) y Numerical results; four-cell convection [3] D case x D Euler A typical non-uniform finite volume cell Ai+,j+ ρv ρuv ρv + p = v (E + p) ρ : density (ρu, ρv )T : momentum vector E : total energy p : pressure = (γ )(E ρ u +v ) Write hyperbolic PDE system as: Q F (Q) F (Q) + + = G(x, y, Q) x y n3 i+,j+ A n i+/,j+/ i,j i,j+ n4 i+,j n

8 Discretization of the grid PDEs A conservative solution-updating method Given a non-uniform partitioning {A i+,j+ } i,j of Ω p, where A i+,j+ is a quadrangle with four vertices x i+k,j+l, k, l (denote numerical approximations to x = x(ξ, η) by x i,j = x(ξ i, η j )) Subdivide Ω c = {(ξ, η) ξ, η } uniformly: (ξ i, η j ) ξ i = i ξ, η j = j η; i I ξ +, j I η + Discretize the elliptic system of grid PDEs by second-order central finite differences Apply a Gauß-Seidel iteration method to the resulting system of algebraic equations Having computed the new grid, the solution values Q have to be updated on this grid by interpolation (Tang et al 3 use a conservative interpolation method) Simple linear interpolation is not good enough Denote with x i,j & x i,j the coordinates of old and new grid points (x i,j moves to x i,j after Gauß-Seidel iterations) Using a perturbation technique and assuming small grid speeds, then the following mass-conservation is satisfied: i,j Ãi+,j+ Q i+,j+ where A is the area of cell A = i,j A i+,j+ Q i+,j+ Finite volume discretization on non-uniform grids [] Finite volume discretization on non-uniform grids [] Q + F (Q) + F (Q) = G(x, y, Q) x y Integration over the finite control volume A i+,j+ : Q dxdy + A i+,j+ 4 F n l (Q) (x,y) sl ds = s l l= A i+,j+ G dxdy Q n+ i+,j+ = Q n i+,j+ t A i+,j+ { F (Q n n i+,j )+F (Q n n i+ 3,j+ ) 4 } +F (Q n n 3 i+,j+ )+F (Q n 3 n 4 i,j+ )+ F + (Q n n l i+,j+ ) + t G n i+,j+ l= The time step size t is determined every time step by with F n l (Q) = F n l x + F n l y and F n l Discretization = F + n l + F n l t = min( x, y) CFL, max λ where λ are the eigenvalues of the acobian matrix F Q

9 D case D Euler Decoupling of the grids and physical PDEs D case D Euler D Riemann problem: shock waves Step Partition Ωc uniformly; give initial partitioning of Ωp ; compute initial grid values by a cell average of control volume Ai+,j+ based on initial data Q(x, y, ). In a loop over the time steps, update grid and solution and evaluate the PDE: Step a Move grid xi,j to x i,j by solving the discretized grid PDEs using Gauß-Seidel iterations. Step b Compute Q i+,j+ based on conservative interpolation. Repeat step a and step b for a fixed number of Gauß-Seidel iterations. Step 3 Evaluate the physical PDEs by the finite volume method on the grid x i,j to obtain Qn+ at t n+. i+,j+ The first test example is a two-dimensional Riemann problem (config. 4 in Lax & Liu 98) and has the following initial data: (.,.,.,.) if x >, y >, (65, 939,., 5) if x <, y >, (ρ, u, v, p)t= = (., 939, 939,.) if x <, y <, (65,., 939, 5) if x >, y <. They correspond to a left forward shock, right backward shock, upper backward shock and a lower forward shock, resp. The spatial domain Ωp is [, ] [, ] and Tend = 5. Step 4 Repeat steps a, b and 3 until t = Tend is reached. D case D Euler Comparison: AL vs. new monitor vs. uniform grid RUN # I II III IV V VI VII VIII IX monitor AL-monitor AL-monitor AL-monitor AL-monitor AL-monitor new monitor new monitor uniform (4 4) uniform (6 6) α m - filter on on on on off off off - runtime h 5m h 47m h 3m 5h 48m 3h 4m h57m h 9m h 5m 3h 5m Table: A shock wave model (configuration 4 from Lax & Liu 98) D case D Euler Numerical results for the AL-monitor left: α =., middle: α =, right: α =

10 D case D Euler Numerical results for the new monitor D case D Euler The adaptive grid: a close-up near (, 5) left: AL-monitor (α = & filter off); middle: AL-monitor (α = & filter on); right: new monitor with m = (no filter needed) left: uniform grid 4 and 6, middle: m =, right: m = D case D Euler The Rayleigh-Taylor instability model? This instability happens on an interface between fluids with different densities when an acceleration is directed from the heavy fluid to the light fluid? It has a fingering nature, with bubbles of light fluid rising into the ambient heavy fluid and spikes of heavy fluid falling into the light fluid? Here we have non-homogeneous BCs and an extra source term in the PDE-system: G = (,, ρg, ρvg )T, where g is the acceleration due to gravity D case D Euler Numerical results for Rayleigh-Taylor instability [] Results at t = and t =.

11 Extension of Winslow to 3D Extension of Winslow to 3D Minimization of grid-energy yields in 3D: with (ω x) = (ω y) = (ω z) =, (x, y, z) T [, ] 3 ω = α(t) + u, α(t) = Ω c u dξdηdζ Minimization of grid-energy yields in 3D: with (ω x) = x τ (ω y) = y τ (ω z) = z τ, (x, y, z)t [, ] 3 ω = α(t) + u, α(t) = Ω c u dξdηdζ Non-singular transformation? Theory? Non-singular transformation? Theory? Transformation of a blow-up PDE models (intro) [] The PDE u = u + up transforms via (x, y, z, t) (ξ, η, ζ, θ); with t = θ to: u θ + [u ξ( x θ (y η z ζ y ζ z η ) y θ (x ζ z η x η z ζ ) z θ (x η y ζ x ζ y η )) + u η ( x θ (y ζ z ξ y ξ z ζ ) Application areas: y θ (x ξ z ζ x ζ z ξ ) z θ (x ζ y ξ x ξ y ζ )) + u ζ ( x θ (y ξ z η y η z ξ ) y θ (x η z ξ x ξ z η ) z θ (x ξ y η x η y ξ ))] = [ (y η z ζ y ζ z η ) + (xζ z η x η z ζ ) + (x η y ζ x ζ y η )! u ξ + ξ «(yη z ζ y ζ z η )(y ζ z ξ y ξ z ζ ) + (x ζ z η x η z ζ )(x ξ z ζ x ζ z ξ ) + (x η y ζ x ζ y η )(x ζ y ξ x ξ y ζ ) u η + ξ «(yη z ζ y ζ z η )(y ξ z η y η z ξ ) + (x ζ z η x η z ζ )(x η z ξ x ξ z η ) + (x η y ζ x ζ y η )(x ξ y η x η y ξ ) u ζ + ξ «(yζ z ξ y ξ z ζ )(y η z ζ y ζ z η ) + (x ξ z ζ x ζ z ξ )(x ζ z η x η z ζ ) + (x ζ y ξ x ξ y ζ )(x η y ζ x ζ y η ) u ξ +... η Combustion models & chemical reaction dynamics Population dynamics: motion of colonies of micro-organisms Plasma physics: wave motion in fluids and electromagnetic fields with = z ζ (x ξ y η x η y ξ ) z η (x ξ y ζ x ζ y ξ ) + z ξ (x η y ζ x ζ y η )

12 models (intro) [] Consider the ODE: Exact solution: { u = up, p > u() = u u(t) = [(p )(T t)] p, T = u p (p ) u(t) models (intro) [3] Consider the PDE: { u = u + up, p > u Ω =, u(x, ) = u (x) Kaplan 963: if u smooth and large enough, then the solution u is regular for every t < T, but lim u(, t) L t T = + u t= t=t solution blows up at time T t models (intro) [4] Adaptive time steps with a Sundman transformation [] Another PDE example: { u = u u + u p, x (, π), t > x u(x, ) = u (x), x (, π), u(, t) = u(π, t) =, t > Let f = π u(x, t) sin(x) dx, then ḟ = π ( u x u p ) sin(x) dx f Via Hölder s inequality: ḟ f + f p p If f () > p p, then f (t) in finite time By Cauchy-Schwartz: f u L (,π) sin(x) L (,π) Explicit Euler with fixed t for { u = up u() = : u n+ = u n + t (u n ) p Numerical solution exists for all t n = n t (i.e. no blow up...) u(t) exact solution num. solution Thus f implies u L (,π) u leaves L (, π) in finite time u t= t=t t

13 Adaptive time steps with a Sundman transformation [] Adaptive time steps with a Sundman transformation [3] Explicit Euler + central FD s for u t = u xx + u p : In terms of scaling invariance and self-similarity: with t n = u n+ j u n j t n θ u n p = un j+ un j + u n j ( x) and constant θ ( a Sundman time transformation t(θ) = θ + (u n j ) p u p for sufficiently small x (+ stab. restriction on t n ), the numerical solution blows up at T x and lim x T x = T (Abia et al, ) ) Consider the ODE u = u, i.e. p =, then using a fictive computational time variable θ gives rise to a new ODE system with { du dθ = u dt dθ = u that is invariant under the scaling t λt, u λ u and for which the numerical solution u n uniformly approximates the true self-similar solution u(t) = t of the original ODE for θ. (Budd, Piggott, Leimkuhler et al) The numerical algorithm in 3D in 3D [] decoupling of blow-up and grid PDEs grid PDEs: system of heat equations with artificial time central finite differences on non-uniform grid for freezing of non-linear terms in PDEs in each time step Case : u(x, y, z, ) = exp( 3((x ) + (y ) + (z ) ) (p = 3, 3 / 3 /4 3 grid; with or without Sundman t(θ)) implicit Euler for and explicit for reaction term BiCGstab + ILU-preconditioning for underlying linear systems variable t using Sundman-transformation Maximum value of u as a function of time t.

14 in 3D [] in 3D [3] Case : Case (cntd.): u(x, y, z, ) = sin(πx) sin(πy) sin(πz) (p = 3, 3 / 3 /4 3 grid; with or without Sundman t(θ)) Adaptive grids and numerical solutions at two points of time just before blow-up (note the scale of the solution in each of the plots) Maximum value of u as a function of time t. (blow-up time in our experiments T from Liang & Lin, 5:.749) in 3D [4] Summary Case (cntd.): D adaptive moving grids: theory and applications! D simple domains : some theory but many applications : lack of theory and still only a few applications Adaptive grid and numerical solution just before blow-up (right plot: close-up of grid and solution)