Implicit multistage two-derivative discontinuous Galerkin schemes for viscous conservation laws

Transcription

1 O C T O B E R P R E P R I N T Implicit multistage two-derivative discontinuous Galerkin scemes for viscous conservation laws Alexander Jaust, Jocen Scütz and David C. Seal Institut für Geometrie und Praktisce Matematik Templergraben 55, Aacen, Germany

2 Implicit multistage two-derivative discontinuous Galerkin scemes for viscous conservation laws Alexander Jaust, Jocen Scütz and David C. Seal October 27, 2015 In tis paper we apply implicit two-derivative multistage time integrators to viscous conservation laws in one and two dimensions. Te one dimensional solver discretizes space wit te classical discontinuous Galerkin DG metod, and te two dimensional solver uses a ybridized discontinuous Galerkin HDG spatial discretization for efficiency. We propose metods tat permit us to construct implicit solvers using eac of tese spatial discretizations, werein a cief difficulty is ow to andle te iger derivatives in time. Te end result is tat te multiderivative time integrator allows us to obtain ig-order accuracy in time wile keeping te number of implicit stages at a minimum. We sow numerical results validating and comparing metods. 1 Introduction In tis work, we focus on viscous conservation laws and present an implicit ig-order time integration scemes for te discontinuous Galerkin DG metod [38, 11, 10, 9, 8, 12]. One of te advantages of using DG is te easy increase of accuracy by locally increasing te polynomial degree wic makes it imperative to also use ig-order time integrators. Explicit Runge-Kutta time integration scemes are often used because of teir low computational costs, low dissipation, favorable stability regions and ease of implementation. However, time step size is limited by te CFL condition wic links bot time step size t and mes size to te speed of propagation of te pysical system. In practical applications, te required time step may be unacceptably small due to very small local mes sizes or te stiffness of te discretized problem. Frequently, a severe CFL restriction can be overcome by using implicit time integrators, allowing larger time steps at te cost of eac time step being more expensive. Te latter is due to te fact tat eac time step requires one or more systems of usually nonlinear equations to be solved. Common examples of implicit time integrators include multistep metods suc as te backward differentiation formulae BDF or multistage metods suc as diagonally implicit Runge-Kutta DIRK metods [1]. One drawback of te discontinuous Galerkin metod witin an implicit solver is te large number of degrees of freedom. Te ybridized discontinuous Galerkin HDG metods, see, e.g., [7, 32, 39], as initially been designed to reduce te memory footprint of steady-state computations, but can also be applied to time-dependent problems [31, 33, 32, 26]. In tis paper, we employ two-derivative metods as our time discretization. As te name suggests, tese algoritms make use of more tan one time derivative and can be constructed to ave a strong stability preserving SSP property [6, 34]. Te principle can be best explained by starting wit Taylor metods, wic are a subclass of te so-called multistage multiderivative metods [20, 40]. Assuming an unknown function wx, t known at some given time t n. Ten, an approximation at some time t > t n could be 1

3 obtained from a Taylor expansion in time wx, t = wx, t n + t t n w t x, t n + t tn 2 w tt x, t n Obviously, witin tis approac, it is necessary to approximate additional time derivatives. In te context of viscous conservation laws tat we define in Eqn. 2, one can replace temporal derivatives by spatial derivatives using te Caucy-Kowalevskaya procedure. In te context of numerical metods, te idea of using Taylor series to discretize time is also often referred to as Lax-Wendroff metod, because tey construct teir second-order sceme using tis procedure [30]. Tis approac is used for so-called Taylor-Galerkin metods from te 1980 s [14, 13], and te original ENO sceme of Harten et al. [22] uses te same procedure. In [15, 16], Dumbser and Munz construct discontinuous Galerkin scemes wit arbitrary order of accuracy in space and time based on ADER scemes. Tey also present an approac to evaluate te Caucy-Kovalevskaya procedure efficiently in [16] based on te work of Dyson [17] relying on te application of te Leibniz rule. Qiu et al. [36] present an approac to couple Lax- Wendroff and discontinuous Galerkin metods LWDG. Tey obtain a ig-order, explicit one step metod for yperbolic problems tat is up to tird order accurate in time. Tey sow tat for teir setting, te LWDG sceme is more efficient tan a Runge-Kutta discontinuous Galerkin RKDG. In [35] te approac is extended to 1D convection-diffusion equations based on te local discontinuous Galerkin LDG metod. Furtermore, te beavior of te metod coupled to different numerical fluxes is studied. Additionally, Taylor discretizations are investigated for finite difference weigted ENO metods in [37, 29, 41, 5]. Seal et. al. [40] are te first to extend te approac to explicit multiderivative Runge-Kutta metods wit DG and WENO spatial discretizations for yperbolic conservation laws in a single dimension. Tey develop a framework for two-derivative Runge-Kutta metods tat can be easily extended to incorporate additional stages or derivatives. Tsai et. al. [44] apply explicit and implicit two-derivative Runge-Kutta metods to PDEs wit ig-order finite-difference metods for spatial discretization. Instead of replacing time derivatives wit spatial derivatives, it is also possible to go te opposite way. Tis is done in te so-called inverse Lax-Wendroff ILW metods [24, 23, 42, 43, 3] In tis paper, we develop a strategy to apply implicit multistep two-derivative metods to convectiondiffusion type equations in 1D using DG. Tat approac is ten extended to first-order PDEs in 2D were we employ HDG for increased efficiency. Te approac sows some similarities to te one in [44], but we use DG instead of finite differences for our spatial discretization. As te time derivatives tat arise from two-derivative time integrators are replaced by spatial derivatives, an accurate way to represent tem is needed. Tis could be done by differentiating te polynomial representation of te solution in te DG setting, aving been justified in [40] by te fact tat te derivatives are multiplied by additional t tat scales as t = O x. In general, te time step restriction for implicit time integration is less severe. Terefore, we employ te LDG approac to accurately represent te additional derivatives, wic as te additional benefit of potentially recovering superconvergence properties [18]. We refer te interested reader to [45] were te application of LDG to PDEs of iger order is discussed extensively. In tis work, we also sow ow to extend our approac efficiently to solve for two dimensional convection problems wit te ybridized DG metod. Te paper is structured as follows. In Section 2, we introduce a 1D nonlinear viscous conservation law tat serves as a model equation. Ten, in Section 3, we briefly describe te two-derivative multistage time integrators tat are used in tis work. Tis also explains te appearance of iger order spatial derivatives tat are not directly present in te underlying PDE. Afterwards, we discretize te model equation in time and space using te LDG approac c.f. Section 4 and verify te metod using linear and nonlinear PDEs in Section 5. Te two-derivative time integration is ten extended to first order PDEs in two dimensions in Section 6. Te resulting equation is discretized using an HDG metod tat significantly reduces te size of te globally coupled system. Finally, we verify te approac using te linear advection and nonlinear Euler equations in Section 7. 2

4 2 Underlying equation In tis work, we begin wit te scalar nonlinear viscous conservation law w t + fw x = εw xx x, t Ω R + 2 wx, 0 = w 0 x x Ω wit ε 0 given on a domain Ω R equipped wit periodic boundary conditions. Te metod to be developed relies - similar to a Lax-Wendroff procedure [30] - on te use of te second temporal derivative w tt, expressed in terms of spatial derivatives. For te underlying problem, we state te following lemma: Lemma 1. Let w C 4 Ω R +. Ten, te second temporal derivative can be expressed as Proof. Obviously, tere olds and consequently, w tt = f wfw x εf ww xx x + ε fw x + εw xx xx =: R 2 w. 3 w t = fw x + εw xx =: Rw w tt = fw x + εw xx t = fw t x + εw t xx = f ww t x + εw t xx = f wfw x εf ww xx x + ε fw x + εw xx xx =: R 2 w. Remark 1 Limiting cases. Te term for w tt simplifies significantly in some limiting cases: 1. If f is linear, i.e., fw = cw, ten 2. Tis also means tat for f 0, w tt = ε 2 w xxxx. 3. If ε 0, tere w tt = f wfw x x. w tt = c 2 w xx 2εcw xxx + ε 2 w xxxx. Note tat te viscous and convective terms influence eac oter mutually, i.e., one obtains cross-terms tat need to be dealt wit. 3 Time integration In tis section, we sortly review multiderivative time integrators as far as it is of importance for tis work. Assume tat te ordinary differential equation y t = gyt is given for a smoot function g. Classical approaces e.g. multistage Runge-Kutta, or linear multistep metods to te numerical approximation of tese equations [19, 21] only use g itself. A multiderivative metod, on te oter and, takes knowledge of iger derivatives of y into consideration. As an example, te second derivative y is given by y t = ġy := g ygy, 3

5 wic can be readily computed for an ODE using symbolic differentiation software. In tis publication, we assume tat 0 t T, and tat tis temporal interval is uniformly subdivided into 0 = t 0 < t 1 <... < t N = T wit spacing t. We note tat none of te algoritms presented depends on a uniform time step size, and tis coice is just for te ease of presentation. As is customary, y n denotes an approximation to y at time t = t n. In tis work, we consider implicit two-point collocation metods wit multiple derivatives of te form m t j D j yy n+1 P m j 0 = j=0 m t j D j yy n P m j 1 4 were P t = tk t 1 l k+l!. Eac of tese metods are of order m = l + k c.f. II.13 in [19]. Because of te growing complexity of iger order derivatives see also Eqn. 3, we rely on scemes involving only two derivatives of te unkown. Remark 2 Employed metods. In tis work, we discuss te tird-order metod wit k = 1, l = 2, given by y n+1 = y n + t 3 and te fourt-order sceme wit k = l = 2 given by y n+1 = y n + t 2 j=0 gy n + 2gy n+1 t2 6 ġyn+1, 5 gy n + gy n+1 + t2 12 ġy n ġy n+1. 6 Tese are te same metods tat are used to discretize te non-linear terms in [4]. More generally, te metods can be written in form y n+1 = y n + t α 1 gy n + α 2 gy n+1 + t 2 β 1 ġy n + β 2 ġy n+1, 7 were te coefficients α i, β i, i = 1, 2 are cosen to maximize te order of accuracy e.g. Eqn. 6, or to modify te region of absolute stability e.g. Eqn. 5 is an L-stable metod. Lemma 2 Stability. Te integrators 5 and 6 are tird- and fourt-order accurate, respectively, and A-stable. Te tird-order metod 5 is L-stable. Applying 7 to equation 2 on a semi-discrete level yields te expression w n+1 tα 2 wt n+1 β 2 t 2 wtt n+1 = w n + tα 1 wt n + β 1 t 2 wtt, n were w tt as to be replaced by te expression in 3. Te term w tt contains spatial derivatives up to fourt order, so we ave to discuss ow to discretize tem in a DG framework efficiently. In [45, 18], te autors sow ow to use te local discontinuous Galerkin LDG metod to discretize te iger spatial derivatives in an explicit DG solver. Teir work will be te basis for te algoritm to be presented in te sequel. 4 1D: Spatial and temporal discretization It is te aim of tis publication to couple temporal discretization in 7 to te discontinuous Galerkin metod. A semi-discretization of 2 is given by w n+1 = w n + t α 1 Rw n + α 2 Rw n+1 + t 2 β 1 R 2 w n + β 2 R 2 w n+1, 8 4

6 were Rw and R 2 w denote te expressions for w t and w tt, respectively c.f. Eqn. 3, and w n denotes an approximation to w at time t n. Before introducing te full spatial and temporal discretization, we start wit some preliminaries. To introduce a finite element metod, we begin by defining a triangulation of Ω into cells Ω k suc tat tey define a partition Ω = N e k=1 wit a total of N e elements. For a given polynomial order p, we define te ansatz space V to consist of cell-wise polynomials of order p wit no continuity restriction along te cell boundaries Ω k V := {q L 2 Ω q Π p Ω k k = 1,..., N e }. Again, it is possible to coose an adaptive p tat differs from cell to cell. exposition. Te metod to be presented relies on te quantities We neglect tis, for ease of σ := u x, τ := σ x = u xx, ψ := τ x = u xxx. In te most straigtforward way, tese variables are discretized as ψ, ϕ Ωk + τ, ϕ x Ωk τ, ϕ n Ωk = 0 ϕ V, τ, ϕ Ωk + σ, ϕ x Ωk σ, ϕ n Ωk = 0 ϕ V, σ, ϕ Ωk + w, ϕ x Ωk ŵ, ϕ n Ωk = 0 ϕ V. As is customary, we ave defined te abbreviations f, g Ωk := fgdx, Ω Ω k k f, g Ωk := Ω k Ω k fgdσx. In one dimension, te last term can be simplified into function evaluations at two points. However, we prefer keeping te integral on te boundary to indicate te algoritm extends to multiple dimensions. Te numerical fluxes τ, σ and ŵ ave to be identified appropriately. One way to acieve a stable sceme is to coose upwinding in an alternating fasion [45]. Te corresponding fluxes read ŵ = w +, σ = σ, τ = τ +, ψ = ψ, 9 were we stick to te convention tat w refers to te left value of w at te interface, and w + rigt value. We summarize tese quantities in an auxiliary variable x V 4 =: X, given by refers to te Tis simplifies te defining equations for σ, τ and ϕ as x := w, σ, τ, ϕ. 10 N aux x, ϕ = 0 ϕ V 3. Remark 3 Lifting operators. It is pointed out in [45] tat it is possible to express te variable σ in terms of w via lifting operators, and subsequently τ and ψ as terms of w as well. Tis comes at te expense of computing lifting operators c.f. [2]. 5

7 Wit tese preliminaries, we now consider te semi-discretization 8 once again. It is well-known ow to spatially discretize Rw using te DG metod Rw, ϕ Ωk nrx, ϕ := fw εσ, ϕ x Ωk fw +, w ε σ, ϕ n Ω k wit te discretization of σ and σ as before. Te numerical flux f denotes a standard consistent and conservative Riemann solver. Details on te cosen flux are given in te numerical results section. Te discretization of R 2 w is less straigtforward. In particular, bot te occurring iger derivatives and te nonlinearity of f pose severe problems. Based on te definition of ψ, τ and σ earlier, we propose te following discretization: R 2 w, ϕ nrtx Ω k, ϕ := f w 2 σ εf w τ, ϕ x + f ŵ Ω k 2 σ εf ŵ τ, ϕ n Ω k + εd 2 f, ϕ x Ω εd 2 k f, ϕ x Ωk ε 2 ψ, ϕ ε 2 ψ, ϕ n Again, te fluxes ŵ, σ, τ and ψ are te LDG fluxes wit alternating evaluation, see 9. D 2 f denotes an approximation to fw xx, see Remark 4. Remark 4 Discretization of D 2 f. Te suitable discretization of D2 f fw xx depends on te coice of te convective flux f. We sow two prototypical examples: Ω k + 1. Linear equation, i.e., fw = cw. In tis case, fw xx = cw xx, and a suitable coice is Ω k. D 2 f := cτ, D2 f := c τ. 2. Burgers equation, i.e., fw = 1 2 w2. In tis case, fw xx = w 2 x + ww xx. As all occurring derivatives are known explicitly in te algoritm, a suitable approximation is D 2 f := σ2 + wτ, D2 f := σ 2 + ŵ τ. 3. A similar procedure as wit Burgers equation is possible wit any flux function - also for Euler equation. However, te result migt become increasingly complex. Ultimately, tis leads to te formulation of te full algoritm, summarized in te following definition: Definition 1 Numerical metod. Let ϕ = ϕ 1 semi-linear form N be given by were N eq x, ϕ 2 is given by N eq x, ϕ 2 1 t N x, ϕ :=, ϕ2 X wit ϕ 1 V 3. Furtermore, let te N aux x, ϕ 1 N eq x, ϕ 2 := α 1N R x n, ϕ2 + α 2N R x n+1, ϕ 2 + t β 1 N R 2x n, ϕ2 + β 2N R 2x n+1, ϕ 2. Te approximate solution x n+1 = w n+1, σ n+1, τ n+1, ψ n+1 X is given as te solution to te problem 0 w n+1 w n, = N x, ϕ ϕ2 ϕ X. Ω k 6

8 Te following lemma is a straigtforward consequence of bot te order of accuracy of te ODE integrator and te consistency of te underlying DG scemes: Lemma 3 Consistency in time. Te algoritm is consistent wit te order of te temporal integration sceme cosen in 8, i.e., tere olds: 1 t 0 w, t n+1 w, t n, ϕ were q = 3 for integrator 5 and q = 4 for integrator 6. Ω k N x, ϕ = O t q Lemma 4 Conservation. Te algoritm is bot locally and globally conservative if D 2 f is conservative. Proof. Testing wit a piecewise constant test function yields tat te integral of w n+1 only depends on te fluxes over te boundaries. Tis yields local conservation. Noting tat te fluxes are conservative and testing against a constant function yields tat te algoritm is globally conservative. 5 1D: Numerical results In tis section, we present numerical results for te newly developed sceme. In eac case, we demonstrate te optimal order of convergence troug. In all our computations, we use periodic boundary conditions on te unit interval Ω := [0, 1], and compute until a final time of T = 0.5. For te cases involving linear convection, we coose te upwind numerical flux fw +, w := cw, c > 0, wereas for Burgers equation, we use a local Lax-Friedrics flux. Te domain Ω is subdivided into equally spaced intervals wit spacing. As an error measure, we compute te L 2 -error at time T, tat is, we define te error as e := w, T w, T L 2 Ω, were w is te exact and w te approximate solution to te underlying problem. 5.1 Heat equation Te first equation to be considered is te pure eat equation w t = εw xx x, t Ω 0, T wit initial conditions w 0 x = sin2πx, and ε = 0.1. Numerical results for different values of te polynomial order p of te ansatz space are sown in Fig. 1 for te tird-order integrator 5 left and te fourt-order integrator 6 rigt. Te expected order of accuracy of max{p + 1, 3} and max{p + 1, 4}, respectively, is acieved. Te time step is set to t = x. Experiments wit oter ratios of t x introduce no stability problems, wic is independent on te coice of ε. Tus, we conjecture tat te algoritm is uniformly stable for tis simple 1D test case witout transport. 7

9 Error e Error e Figure 1: Numerical results for te eat equation. In bot computations, we coose te ratio t x to be one. Temporal integration is performed via te tird-order accurate integrator 5 left and te fourt-order integrator 6 rigt. 5.2 Convection equation Next, we test te algoritm on te pure convection equation w t + cw x = 0 x, t Ω 0, T, again wit initial conditions w 0 x = sin2πx, and constant c = 1. Numerical results are displayed in Fig. 2, again for te tird-order left and te fourt-order rigt temporal integrator. Te CFL number for tis example is one for te tird-order integrator, and only 0.1 for te fourt-order integrator. Te reason for tis coice is tat we find stability constraints wit te fourt-order integrator. Our experience wit oter time integrators as lead to tis in te past, and we suspect tat it is most likely due to te loss of L stability in te fourt-order solver. Numerical experiments indicate tat te tird-order integrator is uniformly stable. We note tat obviously, suc a severe CFL restriction is not a desired feature of an implicit sceme, and a detailed investigation into ow to fix te fourt-order sceme is te subject of future work Error e 10 6 Error e Figure 2: Numerical results for te convection equation. Temporal integration is done via te tird-order integrator 5 left and te fourt-order integrator 6 rigt. Te ratio t x is set to be 1.0 left and 0.1 rigt. Te tird-order integrator seems to be uniformly stable, yet te fourt-order integrator is not, wic is wy we find it necessary to reduce te CFL number. 8

10 5.3 Convection-diffusion equation Te final linear single-dimensional test case is te convection-diffusion equation w t + cw x = εw xx x, t Ω 0, T, wit values c = 1 and ε = 0.1. Tis exercises te ability of te sceme to correctly account for te additional coupling terms tat arise in te discretization of w tt. We present two examples: a an example wit a smoot initial profile, and b a problem wit discontinuous initial conditions Convection-diffusion: Smoot initial conditions. We present numerical results for tis problem in Fig. 3. Te initial conditions for tese results are given by w 0 x = sin2πx, and te ratio t x is cosen to be be one for all cases. For te tird-order integrator, we do not observe any stability issues, wile obviously, from our experiences wit te convection equation, te fourt-order sceme tends to be only conditionally stable. However, tis stability condition does not seem to depend on te magnitude of te viscosity coefficient ε, but rater on te ratio c ε Error e 10 6 Error e Figure 3: Numerical results for te convection-diffusion equation. In bot computations, we coose te ratio t x to be one. Temporal integration is done via te tird-order integrator 5 left and te fourt-order integrator 6 rigt Convection-diffusion: Discontinuous initial conditions. In tis section, we consider discontinuous initial data w 0 x = Hsin2πx 0.3e sin2πx, were H denotes te Heaviside function. Because te initial conditions are not smoot, tere are at least two ways of defining te initial conditions for te auxiliary variables x 0 in Eqn. 10 tat require te spatial derivatives of te initial conditions. One way is to fill it wit te given initial conditions, i.e., set σ 0 = Π V w 0... were Π V denotes te L 2 projection onto V. We use tis coice for te computations in te previous sections, because te initial datum is smoot. However, for tis problem tat as non-smoot initial conditions, w 0 does not exist. For tis case, one alternative is to compute σ 0, τ 0 and ψ0 as a solution to N auxx 0, ϕ = 0 for all ϕ V 3 for a given w = Π V w 0. Tis is te ansatz we pursue in tis section. In Fig. 4, we sow an approximate solution at time T = 0.5, wic uses a spatial widt of = 1 16, quadratic tird-order polynomials, CFL number of t x = 0.5 and te tird-order ODE integrator. We observe a strong agreement between te exact and approximate solution. 9

11 2 w, w, T w, T x x Figure 4: Approximate versus exact solution at time T = 0.5 rigt to te convection-diffusion equation wit discontinuous initial datum left. Results are computed wit, = 1 16 and te tirdorder integrator Viscous Burgers equation Our final single dimensional numerical result is te nonlinear Burgers equation w t + fw x = εw xx wit fw = 0.5w 2 and ε = 0.1. Equipped wit initial conditions w 0 x = sin2πx, tis test case as a smoot solution w for all times T. As before, we sow convergence results in Fig. 5. Te exact solution is computed using te Cole-Hopf transformation [25]. No stability issues are observed in te computations, and te plots sow perfect order of convergence. Te results are similar to tose of te convection-diffusion equation, wic is mainly because diffusion is dominant in tis test case. For implicit metods, tis is probably te most relevant case, as for purely yperbolic problems, explicit metods are often times te preferred metod of coice given te finite speed of propagation of information. Furtermore, we do not implement any limiters but a final algoritm sould ave a suitable way of stabilizing discontinuities in te case of a convection dominated problem. Tis is one topic of future researc, were one option is to introduce artificial viscosity into te time stepping, because we already ave access to tese iger derivatives Error e 10 6 Error e Figure 5: Numerical results for Burgers equation. In bot computations, we coose te ratio t x to be one. Temporal integration is done via te tird-order integrator 5 left and te fourt-order integrator 6 rigt. 10

12 6 2D: Extensions to multiple dimensions In tis section, we describe te extension of implicit two-derivative multistage metods to yperbolic systems of first-order PDEs w t + fw = 0 x, t Ω R + 11 wx, 0 = w 0 x x Ω on a domain Ω R 2 wit appropriate boundary conditions. In general, te unknown w is a function of space and time w := wx, t, but we may drop x and t for a more compact notation. We allow fw to be a general, possibly nonlinear, flux. Note, tat te flux fw as well as w is a vector in R m for a system wit a total of m equations. In order to limit te complexity we stay wit first-order PDEs for te 2D case. Due to te additional spatial dimension, te total number of derivatives also doubles wic would make te assembly of te matrices required for second or iger order PDEs tedious, especially in te case of a system of equations. Neverteless, te two-derivative metods are still applicable to iger order PDEs in 2D using te tecniques from Section 4. Here, we apply te same tird and fourt-order two derivative metods defined in Eqns. 5-6, but we discretize te resulting system using te ybridized discontinuous Galerkin [7, 32, 39] metod. For Poisson s equation, tis discretization is equivalent to te LDG metod [7]. A preliminary investigation tat couples two-derivative Runge-Kutta metods wit HDG for a linear advection equation can be found in in [27]. Te HDG metod usually leads to a muc smaller system of globally coupled equations tan te LDG approac. Tis is especially beneficial in te current case were additional unknowns are introduced by te spatial derivatives from te two-derivative time discretization. We aim to stress at tis point tat te additional unknowns are, for te HDG metod, only local unknowns, and te globally coupled degrees of freedom are not affected. Te representations of Rw and R 2 w only differ sligtly from te one dimensional DG case. Here, we find tat te first and second derivatives are given by w t = fw =: Rw w tt = f w fw =: R 2 w 12 wic follows directly from Eqn. 3 by setting ε = 0. We follow te discretization procedure of Section 4. Te semi-discrete system again reads w n+1 = w n + t α 1 Rw n + α 2 Rw n+1 + t 2 β 1 R 2 w n + β 2 R 2 w n+1 wit Rw and R 2 w defined in Eqn. 12. For te spatial discretization we triangulate te domain suc tat Ω = N e k=1 Te ybridized DG metod requires a description of te edges. We refer to edges of two intersecting elements and elements intersecting te domain boundary Ω wit e k. Te set of all edges is Γ and its number of elements is N f := Γ. Tis is needed to introduce a new ybrid unknown λ = w Γ tat represents te solution evaluated on te trace of eac element. Tis allows us to reduce te size of te globally coupled system by using static condensation [7]. For te approximation of λ λ we need to introduce te ansatz space M tat consists of edge-wise polynomials of degree p defined by Ω k. M := {q L 2 Γ q ek Π p e k k = 1,..., N f, e k Γ} m. 11

13 For te approximation of second order spatial derivatives introduced by a two-derivative time discretization, we again define an auxiliary variable troug σ := w. Te ansatz spaces for σ and w are te common spaces H := {q L 2 Ω q Ωk Π p Ω k k = 1,..., N e } 2m, V := {q L 2 Ω q Ωk Π p Ω k k = 1,..., N e } m, tat contain all polynomials of degree at most p. In order to condense notation, we define te vector of unknowns as x := w, σ, λ tat come from te ansatz space X := H V M, and corresponding test functions ϕ = ϕ 1, ϕ 2, ϕ 3 X. Ten, σ is approximated troug σ, ϕ 1 + w, ϕ 1 λ, ϕ Ω k Ω 1 n = 0 ϕ 1 H, k Ω k tat is very similar to te approximation before, but we use te ybrid variable λ as te numerical flux ŵ := λ. Te equation is abbreviated by N aux x, ϕ 1 = 0 ϕ1 H. Finally, te discretization of R and R 2 for tis first-order PDE is given by Rw, ϕ 2 fw, ϕ 2 and were R 2 w, ϕ 2 Ω k N R x, ϕ 2 := N R 2x, ϕ 2 Ω k := D fw, σ, ϕ 2 Ω k + Ω k 2, f, ϕ n Ω k D f, ϕ 2, n, Ω k f w fw = f wf iw xi w f w f iw σ,i =: D fw, σ. Tus, σ is involved wen te flux is evaluated. On eac element interface, we insert numerical fluxes f = fλ + ηw λ n D f = D fλ, σ θw λ n, tat are modified Lax-Friedrics Rusanov fluxes wit η and θ being stabilization parameters. Wenever θ is multiplied wit a negative coefficient from te time integrator, we invert te sign. Note, tat at tis point te equations are only coupled troug te ybrid variable λ. An additional equation arises from te additional unknown λ troug f + D f, ϕ 3 Ω k = 0 ϕ 3 M. Wit tese preliminaries in place, we are now ready to define te full ybridized DG metod. 12

14 Definition 2 HDG metod. Let ϕ = X. Furtermore, let te semi-linear form N be given by were N eq x, ϕ 2 is given by N eq x, ϕ 2 N aux x, ϕ 1 N x, ϕ := N eq x, ϕ 2, N yb x, ϕ 3 :=α 1N R x n, ϕ2 + α 2N R x n+1, ϕ 2 + t β 1 N R 2x n, ϕ2 + β 2N R 2x n+1, ϕ 2, and te ybrid term is given by N yb x, ϕ 3 := α 2 f n+1 + β 2 D f n+1, ϕ 3 Te brackets denote te jump operator wit vx ± being v = v n v + n vx ± = lim ɛ 0 vx ± ɛn, x Ω k 13 were n is te outward pointing normal. Te approximate solution x n+1 = σ n+1, w n+1, λ n+1 X is given as te solution to te problem 1 t w n+1 0 w n, ϕ2 0 Ω k Γ. = N x, ϕ ϕ X. Remark 5 Number of unknowns. All evaluations on elements only depend on local values of w and σ, and terefore te total number of unkowns can be significantly reduced wen compared to a classical LDG metod. Te coupling between elements is acieved solely by te ybrid variable λ, and terefore, te system to be solved for globally can be condensed [7]. Tis means tat te resulting system is usually muc smaller tan it would be for te standard LDG approac, wic typically requires solving simple local problems on eac element in an element-wise fasion. 7 2D: Numerical results In tis section we sow two-dimensional numerical results. Here, we solve te nonlinear system of equations using Newton s metod. Te resulting linear system is solved using GMRES wit block Jacobi preconditioning until te relative residual drops below Newton s metod is carried out until te L 2 -norm of te residual drops below Linear advection equation We first examine a system of linear advection equations. It can be written as in Eqn. 11. We compute a solution on Ω = [0, 2] 2 at final time T = 0.1. Te flux is cosen to be fw = f 1, f 2 wit f 1 w = A 1 w, f 2 w = A 2 w, 14 13

15 Te vector of unknowns is w = w 1, w 2 T. Te matrices for tis linear system are given by A 1 = , A 2 = Tese matrices ave te same eigenvector basis, wic means we can express tese as A 1 = SD A1 S 1 and A 2 = SD A2 S 1 wit D A1 =, D A2 =, S =, S = After coosing te initial conditions to be w 0 x = sinπx + y, 17 sinπx + y and taking into account periodic boundary conditions, we write te exact solution as sinπx + y + t wx, t =. 18 sinπx + y + t We compute solutions on meses tat are presented in Fig Results are presented in Fig. 7 for te Figure 6: Left : Initial mes. Rigt: Mes after a total of tree refinements. ratio t x = Te errors for w 1 and w 2 are perfectly identical. Te tird order integrator reaces te expected order of convergence in all cases. For, te metod is still tird-order accurate, but it as a lower error tan in te case wit. Te fourt-order integrator, owever, does not acieve fourt-order in time. In te case p < 3, te metod gets close to te expected order of p + 1 wile for te order deteriorates during te refinements. After te sixt refinement it seems not to converge any furter. Most likely, tis is beavior is observed due to stability issues of te fourt-order integrator Euler equations As second test case in two space dimensions, we solve te Euler equations wit periodic boundary conditions. Te flux fw = f 1, f 2 is nonlinear and is given by f 1 w = ρu, P + ρu 2, ρuv, ue + P T, f 2 w = ρu, ρuv, P + ρv 2, ve + P T, 19 14

16 Error e Error e Error e Error e Figure 7: Numerical results for te linear coupled advection equation. Temporal integration is performed wit te tird-order integrator 5 left and te fourt-order 6 integrator rigt. We sow te results for components w 1 top and w 2 bottom. In all computations we coose te ratio t x to be to ensure stability of te numerical metod. and te vector of unknowns is w = ρ, ρu, ρv, E, wic define te density ρ, momentum ρu and ρv in te x- and y-direction, and energy E. Te pressure P is given by te equation of state P = γ 1 E 1 2 ρu2 + v 2, and te ratio of specific eats is γ = 1.4 for te test cases considered in tis work. To analyze te accuracy of te metod, we make use of a test case similar to te one presented in [28] tat as a smoot analytical solution. Te domain Ω = [0, 2] 2 is equipped wit periodic boundary conditions, and te initial conditions are ρx, y = sinπx + y, u = 0.7, v = 0.3, P = A convergence study is presented in Fig. 8, were we compute te solution to a final time of T = 0.5. Bot integrators produce very similar errors, but te tird-order integrator as sligtly lower errors. Te conclusion is tat te iger-order integrator does not exibit any serious advantage over te lower-order integrator for tis test case. For tis problem, we find tat increasing te polynomial order always increases te rate of convergence, wic is in contrast to te previous cases. For example, in te linear test case, going from to decreased te error level, wereas te slope of te error grap stayed almost constant c.f. Fig. 7. For tis problem, tis actually increases te slope. Neverteless, bot integrators ave a sligt loss of convergence rate during refinements. 8 Conclusions and outlook In tis work, we present a novel application of ig-order, implicit multiderivative time integrators to te discontinuous Galerkin framework. Two dimensional results are realized by employing te ybridized 15

17 Error e 10 4 Error e Figure 8: Numerical results for te Euler equations. Temporal integration is done via te tird-order integrator 5 left and te fourt-order 6 integrator rigt. We sow te error in te density ρ. In all computations we coose te ratio t x to be 0.05 to ensure stability of te metods. discontinuous Galerkin metod in order to reduce te total number of unknowns tat would oterwise be required to discretize te system. Results for a tird- and fourt-order time integrator are presented, were we observe te expected order of convergence in time for all of our 1D test cases. However, we numerically observe tat te time integrators are not uniformly stable for convection dominated problems, wic is not a desirable property for an implicit sceme. Tis beavior can also be observed in a more strict sense in te 2D case, were we additionally observe a loss of accuracy for te fourt-order integrator. One possible explanation for tis drawback is tat iger derivatives carry negative coefficients, and effectively introduce anti-diffusion into te metod. Future work must deal wit improving stability properties of tese metods. References [1] R. Alexander. Diagonally implicit Runge-Kutta metods for stiff O.D.E. s. SIAM J. Numer. Anal, 14: , [2] Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini. Unified analysis of discontinuous Galerkin metods for elliptic problems. SIAM Journal of Numerical Analysis, 39: , [3] J. Banks and W. Hensaw. Upwind scemes for te wave equation in second-order form. Journal of Computational Pysics, 231: , [4] M. F. Causley, H. Co, A. J. Cristlieb, and D.C. Seal. Metod of lines transpose: Hig order L-stable ON scemes for parabolic equations using successive convolution. arxiv preprint arxiv: , [5] Andrew J. Cristlieb, Xiao Feng, David C. Seal, and Qi Tang. A ig-order positivity-preserving single-stage single-step metod for te ideal magnetoydrodynamic equations. arxiv preprint arxiv: , [6] Andrew J. Cristlieb, Sigal Gottlieb, Zacary J. Grant, and David C Seal. Explicit strong stability preserving multistage two-derivative time-stepping scemes. arxiv preprint arxiv: , [7] B. Cockburn and J. Gopalakrisnan. A caracterization of ybridized mixed metods for second order elliptic problems. SIAM Journal on Numerical Analysis, 42: ,

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