Sváček, P.

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2022 November 7 - 9, 2022Srní37   conference with international participation  thOn algebraic flux corrections for finite elementapproximation of transport phenomenaP. SváčekCzech Technical University in Prague, Faculty of Mechanical Engineering, Department of Technical Mathematics, Czech Republic1. IntroductionIn this paper we shall focus on numerical discretization of a general hyperbolic system writtenfor the vector of conservative variables u = u(x, t) in the form∂u∂t+∇ · f(u) = 0 in Ω× IT , (1)where f = f(u) denotes the array of the inviscid fluxes. Eq. (1) is equipped with the initialcondition u(x, 0) = u0(x) for x ∈ Ω.For the hyperbolic system (1) the aspects of spatial discretization by the finite elementmethod is discussed. For such a systems there are several physical constraints for the solu-tion as boundedness of physical quantities (e.g., positive density) which needs to be guaranteedalso for the numerical solution. Moreover, the solution of such system needs to satisfy the en-tropy inequality, see, e.g., [1, 6]. The numerical analysis of the approximate method usuallyfocus on the consistency, stability and convergence. The mostly used techniques as finite vol-umes or discontinuous-Galerkin finite elements moreover satisfy the conservativity naturallyby the construction of the scheme. However, for the higher-order finite element method theconservativity needs to be discussed, see, e.g., [2].Moreover, the construction of the numerical scheme should also avoid the occurrence ofnonphysical states. To this end many modern high-resolution schemes use limiters to ensurepreservation of local bounds or at least positivity for scalar quantities of interest. In the con-text of finite element approximations, such schemes can be constructed using the framework ofalgebraic flux correction (AFC) and its extensions to hyperbolic systems, see, e.g., [2,3]. How-ever, the bound-preserving schemes for nonlinear hyperbolic problems are usually not entropystable and vice versa, the entropy stable schemes are usually not bound-preserving. In [4], theAFC scheme is extended for continuous finite element discretization of a scalar conservationlaw using a bound-preserving flux limiter and a semi-discrete entropy fix based on Tadmor’scondition.In this paper the construction of the AFC for continuous linear finite element method isdiscussed, its application realized on a simple linear scalar problem. The attention is paid onthe realization of the Dirichlet boundary conditions. Numerical results are shown.2. FE discretization and its propertiesWe consider a scalar transport equation in the form∂u∂t+∇ · f(u) = s (2)equipped with a suitable initial and boundary conditions. We consider the fluxes in the formf = vu− ε∇u, where v is the velocity and ε ≥ 0 denotes the diffusion.2.1 Linear FE methodFor the purpose of the application of FE method, the weak formulation of (2) is derived in theform ∫Ωw∂u∂t+ w∇ · (vu) +∇w · (ε∇u)dx =∫Ωwsdx+∫ΓNw(ε∇u) · ndS. (3)Consider the FE space Vh with base φj and approximate the solution u and the velocity vby the FE approximations uh and vh as u ≈ uh(x, t) =∑j uj(t)φj(x), v ≈ vh =∑j vjφj .For the multiplication uv the approximation the group FE formulation is used vu ≈ (vu)h =∑j(ujvj)φj. This leads to system of ODE written as∑j(mijduidt)+∑j((cij + dij)uj) = ri (4)where M = (mij) is the mass matrix, C = (cij) represents the convective terms, D = (dij)corresponds to the diffusion terms (proportional to ε) and (ri) are the source terms. For theinviscid limit ε → 0+ with zero source terms we get the system in the form M~̇u = K~u withK = −C, or in the discrete form ∑j(mijdujdt)=∑jkijuj. (5)2.2 Discrete operator propertiesFor the discrete operators the partition of unity(PU) valid for finite elements is important. Tak-ing the partition of unity propery, i.e.,∑j φj = 1, we by differentiation get∑j∇φj = 0.This has direct influence on the properties of the discrete (e.g., mass, Laplace) operators. Themass matrix is symmetric, positive definite and with the PU we have∑i∑j mij = |Ω|. Thediffusion matrix D = (dij) is symmetric with zero row and column sums. The discrete gradi-ent/divergence operator C = (cij) is nonsymmetric with zero row sums whereas the columnsums does not have to be always zero as it is influenced by the mesh properties and boundaryfluxes. For the interior nodes cii = 0 and cij = −cji.2.3 Algebraic flux limiting techniqueThe description of the main idea is shown for the finite element approximation of the problemwith zero viscosity and zero sources, which leads to the discrete equations in the form M~̇u =K~u, where ~u = (ui) is vector of the nodal values, M is the consistent mass matrix and K isthe discrete transport operator. In order to obtain scheme both without undershoots/overshootsas well as not too diffusive, we need to switch between linear “upwind-like” approximationsand the original scheme. In the finite element context the idea of algebraic flux correctionsreads: replace the consistent mass matrix MC by lumped mass matrix ML, and add an artificialdiffusion operator D to operator K to eliminate all negative off-diagonal coefficients of K. Thelinear local extremum diminishing scheme then reads ML~̇u = L~u, L = K + D. The artificialdiffusion operator D can be rewritten as(D~u)i = −∑j 6=ifdij, fdij = dij(ui − uj) = −fdji.The original scheme can be then recovered ML~̇u = L~u − D~u + (ML −MC)~u, or componentby componentmiu̇i =∑jlijUj +∑j 6=ifij, fij = fdij +mij(u̇i − u̇j) = −fji, (6)where mi are coefficients of the lumped mass matrix. In order to prevent the oscillations of thesolution, the fluxes fij are multiplied by suitable correction factorsf ∗ij = αijfij, where 0 ≤ αij ≤ 1.Inserting these fluxes into (6) we get the nonlinear combination of the low order scheme (αij =0) and the original higher order scheme (αij = 1). Following the detailed description from [2]the positive and negative edge contribution to fluxes are accounted for separatelyPi = P+i + P−i , P±i =∑j 6=i min{0, kij}minmax{0, uj − ui},Qi = Q+i +Q−i , Q±i =∑j 6=i min{0, kij}maxmin{0, uj − ui}and use in order to limit positive or negative antidiffusive fluxes. The nodal corrections factorsare computed by R±i = min{1, Q±i /P±i } which determine the percentage of P±i that can beaccepted to node i without violating the LED constraint for row i of the modified transportoperator Kmod. The corrections αij are then computed using a suitable limiter, see [2], byαij ={R+i dij(ui − uj) if ui ≥ uj,R−i dij(ui − uj) if ui < uj.(7)3. Algebraic flux corrections for hyperbolic systemsAs an example we can consider Euler equations in the conservative form ∂W∂t+∇ · F(W ) = 0where W = (ρ, ρu, E)T and flux is given as F(W ) = (ρu, ρu⊗ u + pI, Eu + pu)T Here, thetotal energy E is given as sum of internal energy and kinetic energy, i.e. as E = ρe + 12ρ|u|2and the pressure is then computed using additional equation of state, for ideal gas expressed asp = (γ − 1)ρe. In this case the application of AFC needs to take into account the properties ofthe hyperbolic system, see [3].4. Numerical resultsFirst, the developed flux corrected transport scheme was tested for finite element implemen-tation in 1d and 2d (see Figs. 1–2). The exact solution in this case is just transported with aconstant velocity v and no diffusive fluxes are used, i.e., ε = 0. In Fig. 1 the convection of rect-angle (left) and semiellipse (right) is shown, where the dashed line shows the initial condition,the dotted line shows the exact solution, and the solid line shows the numerical approximation.In this case the transport velocity for 1D case was chosen as v = 1 (1D vector) and the compu-tations was performed for time period of T = 0.5. For the two dimensional case, the convectionof the block with the rotational flow velocity v(x, y) = (−(y − 1/2), (x − 1/2)) around theorigin is approximated. Here, the time period of T = 2π, so that the initial condition is theexact solution at time instant t = T . Fig. 2 shows the exact solution (left) and the numerical0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.910 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91Fig. 1. Algebraic flux corrections (1d)00.5100.5100.20.40.60.8100.5100.5100.20.40.60.81Fig. 2. Algebraic flux corrections (2d)solution(right). Although the solution is slightly smeared, no undershoots or overshoots weredetected.AcknowledgementsAuthors acknowledge support from CTU and computational time provided by the Center ofAdvanced Aerospace Technology (CZ.02.1.01/0.0/0.0/16 019/0000826), Faculty of Mechani-cal Engineering, Czech Technical University in Prague.References[1] Dafermos, C. M., Hyperbolic conservation laws in continuum physics, 1st edition, Springer, 2000.[2] Kuzmin, D., Möller, M., Algebraic flux correction I. Scalar conservation laws, In: Flux-correctedtransport, Kuzmin, D., Löhner, R., Turek, S. (eds), Scientific Computation, Springer, Berlin, 2005.[3] Kuzmin, D., Möller, M., Algebraic flux correction II. Compressible Euler equations, In: Flux-corrected transport, Kuzmin, D., Löhner, R., Turek, S. (eds), Scientific Computation, Springer,Berlin, 2005.[4] Kuzmin, D., Quezada de Luna, M., Algebraic entropy fixes and convex limiting for continuous fi-nite element discretizations of scalar hyperbolic conservation laws, Computer Methods in AppliedMechanics and Engineering 372 (2020) No. 113370.[5] Kuzmin, D., Turek, S., Flux correction tools for finite elements, Journal of Computational Physics175 (2002) 525–558.[6] LeVeque, R. J., Numerical methods for conservation laws, Birkhäuser, 1992.

On algebraic flux corrections for finite element approximation of transport phenomena

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On algebraic flux corrections for finite element approximation of transport phenomena

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