Multiscale hp-adaptive quiescent reactors

Multiscale hp-adaptive quiescent reactors C.E.Michoski Institute for Computational Engineering and Sciences (ICES) University of Texas (UT) at Austin Group Meeting Nov. 7, 2011

What are quiescent reactors? Reactors with no mixing

What are quiescent reactors? Generally concerned with systems (i.e. n > 1 species) of reaction-diffusion equations, which are nonlinear in the reaction term. Such systems admit discontinous solutions, with physical meaning, such as phase-changes, action-potentials, oscillatory chemical reactions,... In spatial dimension N > 2, the Hausdorff dimension of the singularity set for quadratic reactions has a known upper bound dim H V N 2 4/N. Not all reaction-diffusion systems satisfy the requisite underlying assumptions, leading to so-called unphysical solutions in the physicist s lingo.

Outline 1 Equation (or normal) ordering Conservation laws 2 Equation (or normal) ordering Conservation laws 3 Fractional multistep operator splitting Law of mass action Mass diffusivty 4 The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability 5

Where does it come from? Equation (or normal) ordering Conservation laws The idea is that physical systems must obey a normal ordering of the physics, so that each system must be a direct consequence of a more fundamental system. If not, the system gets labelled unphysical. If so, we say it follows from first principles. e.g. QFT Standard Model Quantum Mechanics Statistical mechanics Continuum Mechanics Macroscopic continuum e.g. abstract mathematics canonical quantization KG-Dirac equations Schrödinger equation Boltzmann Equation Navier-Stokes equations Shallow Water equations

The species Boltzmann equations Equation (or normal) ordering Conservation laws Consider a system of n PDEs in Ω Ω (0, T ), where Ω R N such that: t f i + v i x f i = S i (f ) + C i (f ). in position x R N, species velocity v i R N, and time t (0, T ) over index i = 1,..., n. Distribution functions: n distribution functions f i = f (t, x, v i ) Reactive collisional integral C i = C i (f 1,..., f n ) Scattering collisional integral: S i = S i (f 1,..., f n )

The formal Enskog Expansion Equation (or normal) ordering Conservation laws Then we consider the usual Enskog expansion of to linear order t f i +v i x f i = ɛ 1 S i (f )+ɛ b C i (f ), and f i = fi 0 ( 1 + ɛηi + O(ɛ 2 ) ), where ɛ is the formal expansion parameter. The perturbations η i are used to determine the respective forms of the corresponding transport coefficients. When b = 0 then we are in the so-called strong reaction regime When b = 1 we are in the Maxwellian reaction regime When b = 1 we are in the kinetic chemical equilibrium regime.

Equation (or normal) ordering Conservation laws Approx. representation by continuum mechanics After expanding with respect to the collisional invariants of the species Boltzmann equation (keeping terms of the same order ɛ), we arrive with the species Navier Stokes equations: t ρ i + x (ρ i (u V i )) δ b0 m i A i ( n) = 0, t (ρu) + x (ρu u) + x S = 0, t (ρe) + x ((ρe + S)u) + x Q = 0 Notice that this is not a system of reaction-diffusion equations.

The quiescent reactor constraints Equation (or normal) ordering Conservation laws In order to restrict to a true reaction diffusion system, the following constraints must hold: n (1) C i (f 0 ) f C i (f 0 ) i, (2) v i dv i 0, (3) x ρ i x ϑ i, (4) h i (n i m i pd ij ) 1 θ i. i Ω Constraint (1) is standard and observable at high energy. Constraint (2) assumes that the global flow velocity averages to zero over the entire domain (no appreciable mixing). Constraints (3) and (4) restrict to solutions that constrain the admissible bounds on the thermal gradients

Equation (or normal) ordering Conservation laws (the equations for...) The we arrive with a solution over (t, x) (0, T ) Ω for Ω R N chosen to satisfy: t ρ i x (D i x ρ i ) A i ( n) = 0, n n A i ( n) = m i (νir b νir f ) k fr k br r R with initial-boundary data on Ω, ρ i (t = 0) = ρ i,0, and a 1i ρ i,b + x ρ i,b (a 2i n + a 3i τ ) = a 4i, taking arbitrary functions a ji = a ji (t, x b ) for j {1, 2, 3, 4}, where n is the unit outward normal and τ the unit tangent vector. j=1 n νf jr j j=1 n νb jr j,

Variables and parameters Equation (or normal) ordering Conservation laws n i is the molar concentration of the i-th chemical constituent, which up to a scaling by Avogadro s constant N A is just the number density n i = N A n i. The species densities are given by ρ i = ρα i = m i n i, where α i is the mass fraction of the i-th species, and m i is the molar mass of the i-th species. The D i are the interspecies mass diffusivity coefficients, which have a complicated form. The A i ( n) is the chemical mass action, which depends on the reaction rates k r and the realtive constituents availability ν jr.

The form of chemical equations Equation (or normal) ordering Conservation laws The forward and backward stoichiometric coefficients of elementary reaction r N are given by ν f ir N and νb ir N, while k fr, k br R are the respective forward and backward reaction rates of reaction r. The indexing sets R r and P r as the reactant well and the product well R r N and P r N for reaction r. Then, j R r ν f jr M j k fr k br k P r ν b kr M k, r R. (1)

Conservation principles Equation (or normal) ordering Conservation laws Since the elementary reactions are balanced the conservation of atoms in the system is an immediate consequence of (1). Let a il be the l-th atom of the i-th species M i, where l A r is the indexing set A r = {1, 2,..., n atoms,r } of the distinct atoms present in each reaction r R. Then the total atom conservation is satisfied for every atom in every reaction a il νir f = a il νir b r R, l A r. i R r i P r Since the total number of atoms is conserved, so is the total mass, m i νir f = m i νir b r R. i R r i P r

Conservation principles, cont. Equation (or normal) ordering Conservation laws It then immediately follows that an integration by parts yields the following bulk conservation principle which is satisfied globally: d n ρ i dx = 0. (2) dt i=1 Ω Moreover, the system of chemical reactions satisfy spontaneity G 0 (up to a constant) with respect to the standard total Gibb s free reaction energy of the system, which further provides that the entropy of the system dissipates.

Outline 1 Fractional multistep operator splitting Law of mass action Mass diffusivty 2 Equation (or normal) ordering Conservation laws 3 Fractional multistep operator splitting Law of mass action Mass diffusivty 4 The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability 5

Fractional multistep operator splitting Law of mass action Mass diffusivty Now we consider the multiscale solutions that split over fast ρ f and slow ρ s modes. Assume that ρ = ρ f + ρ s, such that (ρ f ) t x (D f σ f ) A f = 0, and σ f x ρ f = 0, (ρ s ) t x (D s σ s ) A s = 0, and σ s x ρ s = 0. Then the splitting of the reaction part at time t provides { (ρf ) reaction modes t A f = 0, (ρ s ) t A s = 0. and the diffusion part splits as: { (ρf ) diffusion modes t x (D f σ f ) = 0, with σ f x ρ f = 0, (ρ s ) t x (D s σ s ) = 0, with σ s x ρ s = 0.

Composition of modes Fractional multistep operator splitting Law of mass action Mass diffusivty Note that the operation of composition here is given in the natural way, such that (at second order) D t/2 (ρ) R t (ρ) D t/2 (ρ) means we solve the system: (ρ ) t x (Dσ ) = 0, ρ (0) = ρ 0 on [0, t/2] (ρ ) t A = 0, ρ (0) = ρ 0 on [0, t], (ρ ) t x (Dσ ) = 0, ρ (0) = ρ 0 on [0, t/2], where the solution to the composition is then given by ρ (t/2) = ρ (t/2, x).

Consistency of multiscale splitting Fractional multistep operator splitting Law of mass action Mass diffusivty Reaction Regnant Diffusion Limited Transition Regimes Diffusion Dominated Shock Dominated

nfanodes Fractional multistep operator splitting Law of mass action Mass diffusivty The law of mass action splits into an n-coupled system of first order ordinary differential equations (nfanodes), which have no spatial dependence. As such, developing a solution technique generally rests somewhere between: (1) finding a relatively straightforward approximate solution, and (2) analytically solving the difficult (though often soluble) system of nfanodes for a time-evolving solution to the split equation.

Complicated functional parameter Fractional multistep operator splitting Law of mass action Mass diffusivty The Enskog expansion tells us that the mass diffusivity (which obey Fick s second law of diffusion) also must satisfy: where p is the pressure, and D i = ρ i p 1 D ij ρi p, D ij = D ij (p, ϑ, v i, u, x p, x ϑ, x u, x v i ), correspon to the diffusivity matrix. Note, however that matrix diag(ρ i )D with diagonal components ρ i D ij is a symmetric positive semidefinite matrix.

Outline 1 2 Equation (or normal) ordering Conservation laws 3 Fractional multistep operator splitting Law of mass action Mass diffusivty 4 The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability 5 The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability

The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability Take test functions ϕ hp, ς hp, ϖ hp W k,q (Ω h, T h ) characterized by ϕ hp (x) = p ϕ i l Ni l (x), ς hp(x) = l=0 and ϖ hp (x) = p ς i l Ni l (x) l=0 p ϖ i l Ni l (x) x Ω e i, where ϕ i l, ςi l and ϖi l are the nodal values of the test functions in each Ω ei, and with the broken Sobolev space over the partition T h defined by W k,q (Ω h, T h ) = {w : w Ωei W k,q (Ω ei ), w w, Ω ei T h }. l=0

Find the variational form The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability The time derivative terms yield: d ( ) ρhp, ς hp d ( ) dt Ω ρ, ςhp G d Ω G d ( ) and ρhp, ϕ dt hp d ( ) Ω ρ, ϕhp G dt and the mass action term gives, ( Ahp ( n), ϕ hp )Ω G ( A ( n), ϕ hp ) Ω G, and the mass diffusion term, ( x (I σ), ς hp )Ω G = Ω ei T h Ω G, Ω ei x (ς hp I σ)dx ( I σ, x ς hp ) Ω G.

One more integration by parts The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability We approximate the first term on the right in the diffusion using the generalized diffusive flux G ij in the unified sense, such that G i = G i (I hp, σ hp, ρ hp, ς hp ), and we see that G i = j S(i) j S(i) Γ ij Gij (I hp, σ hp Γij, σ hp Γji, ρ hp Γij, ρ hp Γji, n ij ) ς hp Γij dξ Γ ij s=1 N (I hp σ) s (n ij ) s ς hp Γij dξ.

The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability The interior diffusion term is approximated directly by ( ) ( ) H = H (I hp, σ h, ρ hp, ς hp ) = I hp σ hp, ς hp x I x ρ, ς hp x Ω G and the auxiliary numerical flux is taken so that: L i = L i ( L ij, σ hp, ρ hp, ϖ hp, ϖ hp x, n ij ) = ( ) σ hp, ϖ hp Ω ei ( + ρ hp, ϖ hp x )Ωei L (ρhp Γij, ρ hp Γji, ϖ hp Γij, n ij )dξ, where j S(i) Γ ij j S(i) Γ ij Lij (ρ hp Γij, ρ hp Γji, ϖ hp Γij, n ij )dξ j S(i) Γ ij s=1 N (ρ) s (n ij ) s ϖ hp Γij dξ. Ω G,

Semidiscrete formulation The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability For each t > 0, find the pair (ρ hp, σ hp ) such that The semidiscrete discontinuous Galerkin scheme a) ρ hp C 1 ([0, T ); S d h ), σ hp S d h, b) ρ hp (0) = Π hp ρ 0, d ( ) c) ρhp, ς hp = G + H, L = 0, dt Ω G d ( d) ρhp, ϕ dt hp = )Ω ( ) A hp ( n), ϕ G hp. Ω G

RKSSP or RKC discrete form The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability The discrete split explicit RKDG scheme a) ρ hp (0) = Π hp ρ 0, b) ρ n f hp = ρ hp(0), ( ( cd) ρ ns+1 hp, ς hp = )Ω ( G dc) ρ ns+1 hp, ϕ hp = )Ω G ρ (χ) e) Y T = Y T (D, R, t ns+1 ). ) hp, ς hp ( ρ (χ) hp, ϕ hp, L (χ 1) = 0, Ω ) G Ω G.

The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability The point of the RKC scheme is to extend the real stability region of the Jacobians in the operator splitting while maintaining second order consistency (others exist, such as the ROCK and DuUMKA schemes, fourth order consistent),.

IMEX discrete form The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability The discrete split IMEX DG scheme a) ρ hp (0) = Π hp ρ 0, b) ρ n f hp = ρ hp(0), ( ( ) cd) ρ ns+1 hp, ς hp = ρ )Ω (χ) hp, ς hp, L (χ 1) = 0, G Ω ( ( ) G dc) ρ ns+1 hp, ϕ hp = ρ )Ω ns hp, ϕ hp + t ns Z ( ) A hp ( n), ϕ hp, G Ω G e) Y T = Y T (D, R, t ns+1 ).

Implicit schemes for the mass action The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability Here the implicit timestepping in (dc) is chosen such that we implement the usual back differentiation formulas (BDF(k)) of order k. Here we use Newton Krylov methods with low accuracy tolerances such that the explicit diffusion step stability is taken as the stability limiting step. By default the Krylov method used is GMRES, while the Newton iteration is based the standard Jacobian line search methods.

L 1 -stability (TVB) in quiescent reactors The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability Ends up, that the stoichiometric coefficients determine a weighting prefactor, given each reactions equilibrium constant K eq,r = k fr k 1 br, κ i = [ r R ] 1/n( r R (νb ir νf ir )) K eq,r so that the entropy satisfies: { n S R = n i ( ln κ i n i + κ i )dx + + n i=1 sup 0 t T t 0 Ω i=1 Ω n 1 i D i x n i 2 dxds } n i=1 P 0 r R + t 0 Ω m 1 i D( n)dxds n 2(e κ i ) 1 Ω + C(T ). i=1 (3)

p-enrichment via dioristic entropy The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability Now, we use the discrete local-in-time change in entropy density, ( ) ˆρS k+1 R,Ω ei = ˆρ S k+1 R,Ω ei SR,Ω k ei to develop the p-enrichment/de-enrichment strategy, where E k+1 i = ( P s+1 (Ω k+1 e i ) if ˆρS k+1 R,Ω ei ϱs k+1 ) R,Ω s+1 ei µ s+1 (s + 1 p max) (τ 0 t w ), ( P s 1 (Ω k+1 e i ) if ˆρS k+1 R,Ω ei ϱs k+1 ) R,Ω ei s 1 µ s 1 (s 1 p min ) (τ 0 t w ), given the change in the average global entropy density ( ϱs k+1 R = ϱ S k+1 R ) S R k. The notation { } is simply a pair from the indefinite binary relations { } {<, >,, } where the choice determines the difference between a Type I and and Type II enrichment scheme.

The basic feature: a stabilizing center The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability Denoting the global average change in entropy density by Avg ΩG ˆρS k+1 R,Ω ei, the stabilizing center at timestep k + 1 is the discrete subdomain c Ω hp comprised of the union of elements over which the change in entropy density satisfies the condition, } c = { 1 j ne Ω ej : ˆρS k+1 R,Ω ei Avg ΩG ˆρS k+1 R,Ω ei s µ s, i.

The entropic jump and hp-adaptivity The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability The entropic jump (or the entropic flux ) across element boundaries, similaryl lead to the density of the variation in the entropic jump leading to h-adaptivity functional: ( ) ˆρ J k+1 R,Ω ei = ˆρ J k+1 R,Ω ei JR,Ω k ei, ( T A k+1 h (Ω k+1 e i ) if ˆρ J k+1 R,Ω i = ei ϱ J k+1 ) R h η h (s + 1 h max) (τ 0 t w ) ( T h0 (Ω k+1 e i ) if ˆρ J k+1 R,Ω ei ϱ J k+1 ) R h η h0 (s 1 h min ) (τ 0 t w ) where h max and h min correspond to the maximum and minimum refinement levels, the density of the global change in the entropic jump is given such that: ( ϱ J k+1 R = ϱ J k+1 R ) J R k.

Kinetic switch hp-adaptivity functional The spatial discretization The semidiscrete formulation The discrete formulations Entropy enriched hp-adaptivity and stability Finally, we couple the h-adaptivity functional A k+1 i to the p-enrichment functional E k+1 i such that h-adaptivity is always preferentially chosen over p-enrichment. We evaluate the simple kinetic switch functional K k+1 i = K k+1 i (A k+1 i, E k+1 i ) determined by evaluating: A k+1 i E k+1 i if T h (Ω k+1 e i ) P s (Ω k+1 e i ) S k+1 K k+1 i = R, R, R, R, A k+1 i if T h (Ω k+1 e i ) P s+1 (Ω k+1 e i ) S k+1 A k+1 i E k+1 i if T h0 (Ω k+1 e i ) P s (Ω k+1 e i ) S k+1 A k+1 i E k+1 i if T h0 (Ω k+1 e i ) P s+1 (Ω k+1 e i ) S k+1 0 otherwise.

Lotka-Volterra reactor with n = 3 species Any system satisfying: t ρ i x (D i x ρ i ) L i ( n) = 0 L i ( n) = λ i n i + n i n j=1 A ij n k=1 nb jk k } GLV system, where λ i is a real or complex parameter and B and A are real or complex valued square matrices, a generalized Lotka-Volterra (GLV) system.

Then we consider an exact case when n = 3 corresponds to three chemical species, j R r ν f jr M j k fr k br k P r ν b kr M k, r R, that splits over the mass action in its components to satisfy, dρ 1 dt = N 1 n 1 + U 1 n 2 1 + O 12n 1 n 2 + O 13 n 1 n 3, dρ 3 dt r R U i = m i dρ 2 dt = N 3 n 3 + O 31 n 3 n 1 + O 32 n 3 n 2 + U 3 n 2 3, where the constants are given by, ) N i = m i (νir b νf ir (1 ) {ν f ir >0} k fr 1 {ν b ir >0} k br, r R (ν b ir νf ir ) (1 {ν f ir >1} k fr 1 {ν b ir >1} k br = N 2 n 2 + O 21 n 2 n 1 + U 2 n 2 2 + O 23n 2 n 3, ), r R ) O ij = m i (νir b νf ir (1 ) {ν f ir,ν jr f >0}k fr 1 {ν b ir,ν jr b >0}k br,

Reactor with ten reactions (r = 10) and three species (n = 3) Then for the N i s being any excess/bulk/bath constituents, a reactor system comprised of ten coupled reactions that satisfies this equation is: M 1 + M 2 k f1 2M 2, M 1 + M 3 k f2 3M 3, 2M 1 + N 1 k f3 3M 1, M 2 + M 3 + N 2 k f4 2M 2, 2M 2 + N 3 k f5 M 2 + N 4, M 1 + M 3 k f6 3M 1, 2M 3 + N 5 k f7 3M 3, ν f 18 M 1 k f8 N 4, ν f 29 M 2 k f9 N 5, ν f 310 M 3 k f10 N 6. The integrals of motion then yield the following exact time-dependent solutions: n 1 = I 1 N 1 N 2 N 3 e N 1 m 1 1 t I 3 I 1 (I 1 U 1 N 2 N 3 e N 1 m 1 1 t + I 2 U 2 N 1 N 3 e N 2 m 1 2 t + I 1 I 2 U 3 N 1 N 2 e N 3 m 1 3 t ) I 2 N 1 N 2 N 3 e N 2 m 1 2 t n 2 =, I 3 I 1 (I 1 U 1 N 2 N 3 e N 1 m 1 1 t + I 2 U 2 N 1 N 3 e N 2 m 1 2 t + I 1 I 2 U 3 N 1 N 2 e N 3 m 1 3 t ) I 1 I 2 N 1 N 2 N 3 e N 3 m 1 3 t n 3 =, I 3 I 1 (I 1 U 1 N 2 N 3 e N 1 m 1 1 t + I 2 U 2 N 1 N 3 e N 2 m 1 2 t + I 1 I 2 U 3 N 1 N 2 e N 3 m 1 3 t ).

When D 0, we have an analytic solution Figure: The solution ρ 1 in N=2.

When D 0, we have an analytic solution Figure: We plot the p-convergence of ρ1, where in N = 2 we set h/10 = 1/32 and for N = 3 we have h/10 = 1/16.

When D 0, we have an analytic solution, cont. Figure: The h-convergence of ρ1 with N = 2, where the h-levels are defined as the h/10 values here and below.

When D 0, we have an analytic solution, cont. Figure: Here we show the h-convergence of the equilibrium solution, where p = 1 and p = 2 in N = 3. Again the h-levels are the h/10 values.

Adding back nonvanishing mass diffusion, D 0 With vanishing mass diffusion, only p-enrichment occurs. The h-adaptivity occurs along the profile of the change in entropy density over element interiors. The p-enrichment solution satisfies the minimal convergence requirements. With nonzero mass diffusion, h-adaptivity may occur. The h-adaptivity occurs along the profile of the entropic jump over interior faces. The solution satisfies the minimal convergence requirement, with p-enrichment off.

e.g. h-refinement minimal convergence requirement IMEX L 2 -error RKC, ɛ = 2/13, L 2 -error h p = 1 p = 2 p = 3 p = 1 p = 2 p = 3 1/64 0.00026 4.6e-06 0.00033 5.4e-06 1/32 0.00084 3.1e-05 1.4e-06 0.00106 3.9e-05 1.4e-06 1/16 0.00173 0.00031 1.9e-05 0.00219 0.00039 2.4e-05 hp ι s = ι h = 0.5, 2.7e-06 ι s = ι h = 0.5, 3.3e-06 Table: We give the L 2 -errors for the h and p ramping of the equilibrium solution with homogeneous diffusion added. Here we have set t = 0.01 seconds, and T = 0.5 s. The p-enrichment scheme is of Type II and the h-adaptive scheme is of Type I, and both span all h and p levels (i.e. p {1, 2, 3} h {1/16, 1/32, 1/64}).

k+1 The gradient σ h versus the entropic jump ρ JR,Ω h Figure: Here we show the derived variables evaluated at timestep five using t = 0.01 with hp-adaptation turned off, and h = 1/256.

hp-adapted, h-refinement profile Figure: Here we show an example of the h-refinement scheme that emerges after eleven timesteps of t = 0.01, where we have a Type I refinement scheme for h and a Type II enrichment scheme for p.

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