Implicit Multistage Two-Derivative Discontinuous Galerkin Schemes for Viscous Conservation Laws Link Peer-reviewed author version

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1 Implicit Multistage Two-Derivative Discontinuous Galerkin Scemes for Viscous Conservation Laws Link Peer-reviewed autor version Made available by Hasselt University Library in Document Reference Publised version): Jaust, Alexander; Scütz, Jocen & Seal, David C.2016) Implicit Multistage Two-Derivative Discontinuous Galerkin Scemes for Viscous Conservation Laws. In: JOURNAL OF SCIENTIFIC COMPUTING, 692), p DOI: /s x Handle: ttp://dl.andle.net/1942/21525

2 Journal of Scientific Computing manuscript No. will be inserted by te editor) Implicit multistage two-derivative discontinuous Galerkin scemes for viscous conservation laws Alexander Jaust, Jocen Scütz and David C. Seal Received: date / Accepted: date Abstract In tis paper we apply implicit two-derivative multistage time integrators to 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. Keywords discontinuous Galerkin metod multiderivative time integration convection-diffusion equation ybridized discontinuous Galerkin metod 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) me- A. Jaust Vakgroep wiskunde en statistiek, Hasselt University Agoralaan Gebouw D, BE-3590 Diepenbeek, Belgium alexander.jaust@uasselt.be J. Scütz Vakgroep wiskunde en statistiek, Hasselt University Agoralaan Gebouw D, BE-3590 Diepenbeek, Belgium jocen.scuetz@uasselt.be D.C. Seal United States Naval Academy Department of Matematics 121 Blake Road Annapolis, MD USA seal@usna.edu

3 2 Alexander Jaust, Jocen Scütz and David C. Seal tod [11,12,13,14,15,41]. One advantage of DG is tat it is easy to increase spatial accuracy by locally increasing te polynomial degree, owever tis ig-order accuracy is lost unless a ig-order time integrator is applied to te result. For yperbolic problems, 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, te time step size t is limited by Courant-Friedrics-Lewy CFL) stability constraints wic link t to te mes size and te speed of propagation of te pysical system). In practical applications, te maximum allowable time step may be unacceptably small due to very small local mes sizes. Tis becomes especially troublesome for problems wit diffusion, were te maximum speed of propagation of information is infinite. Tese severe CFL restrictions can frequently be overcome by using implicit time integrators, wic may permit for larger time steps at te expense of increased computational cost per time step. Tis is attributed 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 cief criticism of te discontinuous Galerkin metod is te large number of degrees of freedom required to compute a solution. Tis is especially troublesome for te case of time dependent problems. As part of an effort to reduce te memory footprint for steady-state computations, te ybridized discontinuous Galerkin HDG) metod [10, 36, 42] was proposed as an alternative to more classical DG metods. More recently, tis metod as been applied to time-dependent problems [29, 34, 35, 36], wic by construction, requires te use of implicit time integrators. 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 [8, 37]. Te principle can be best explained by starting wit Taylor metods, wic are a subclass of te so-called multistage multiderivative metods [23,43]. If wx, t) is a function wose values are only known at time t = t n, ten approximations at some time t > t n can be 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 ) +. 1) 2 It is necessary to approximate additional time derivatives wit tis type of discretization. In te context of viscous conservation laws tat we define in Eqn. 2)), we replace temporal derivatives by spatial derivatives using te Caucy-Kovalevskaya CK) procedure. In te context of numerical metods for partial differential equations, te use of Taylor series in time) to discretize te PDE is often attributed to Lax and Wendroff [33]. Tere, te autors write down a second-order accurate Taylor series, and ten appeal to te PDE to convert temporal derivatives

4 Multiderivative DG 3 to spatial derivatives. Tis approac is used for te so-called Taylor-Galerkin metods from te 1980 s [16,17], and te original ENO sceme of Harten et al. [25] uses te same procedure. In [18, 19], Dumbser and Munz construct discontinuous Galerkin scemes wit arbitrary order of accuracy in space and time based on Arbitrary DERivative ADER) scemes. Tey also present an approac to evaluate te Caucy-Kovalevskaya procedure efficiently in [19] based on te work of Dyson [20] tat relies on te application of te Leibniz rule. In addition, Qiu et al. [39] present an approac to couple Lax-Wendroff and discontinuous Galerkin metods LWDG) based upon direct differentiation of te basis functions to define iger-order derivatives of te solution. 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 [38] 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 [7,32,40,44]. Seal et. al. [43] are te first to extend te Lax-Wendroff type of 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. In addition, Tsai et. al. [47] apply explicit and implicit two-derivative Runge-Kutta metods to PDEs wit ig-order finite-difference metods for spatial discretization. Finally, we point out tat te applications of te Caucy-Kovalevskaya described tus far are used for temporal evolution of te solution, but te same procedure can also be used to define ig-order boundary values by going te oter way around. Tat is, in place of using te PDE to replace time derivatives wit spatial derivatives, it is possible to define spatial derivatives e.g., normal derivatives of te solution along te boundary of a domain) from time derivatives of te solution. Tis is done in te so-called inverse Lax- Wendroff ILW) metods [27,45,46], as well as oter related works [3,26]. Our focus is not on tis application, but on making use of te CK procedure to define te temporal evolution for te solution. In tis paper, we develop a strategy to apply implicit multistep twoderivative metods to convection-diffusion 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 [47], 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. Tis approac is justified in [43], wic explains tat te derivatives are multiplied by additional powers of t tat scale like t = O x). In general, te time step restriction for implicit time integration

5 4 Alexander Jaust, Jocen Scütz and David C. Seal 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 [21]. We refer te interested reader to [48], were te application of LDG to PDEs of iger order is discussed extensively. In tis work, we also sow ow to extend our approac to efficiently solve for two dimensional convection problems wit te ybridized DG metod. Te remainder of tis 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 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 [33] - 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 w tt = f w)fw) x εf w)w xx ) x + ε fw) x + εw xx ) xx =: R 2 w). 3) Proof Obviously, tere olds and consequently, w t = fw) x + εw xx =: Rw) w tt = fw) x + εw xx ) t = fw) t ) x + εw t ) xx = f w)w t ) x + εw t ) xx = f w)fw) x εf w)w xx ) x + ε fw) x + εw xx ) xx =: R 2 w).

6 Multiderivative DG 5 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 w tt = c 2 w xx 2εcw xxx + ε 2 w xxxx. 2. Tis also means tat for f 0, w tt = ε 2 w xxxx. 3. If ε 0, tere w tt = f w)fw) x ) x. 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. As is customary, te prime denotes differentiation wit respect to t.) Classical approaces e.g., multistage Runge-Kutta, or linear multistep Adams metods) to te numerical approximation of tese equations [22, 24] 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) = g y) t) := y gy) y t) = y gy) gyt)), 4) wic can be readily computed for a system of ODEs 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 in tis work depend on a uniform time step size; tis coice is simply made for te ease of presentation. As is customary, y n denotes an approximation to y at time t = t n, 0 n N. Te metods considered in tis work are implicit two-point collocation metods tat make use of multiple derivatives of te solution tat take te form m m t j j t y)y n+1 )P m j) 0) = t j j t y)y n )P m j) 1), 5) j=0 were P t) = tk t 1) l k+l)! and j t is te j t temporal derivative of y. Tese metods can be found by fitting a Hermite-Birkoff interpolation tat matces a total of k derivatives of te solution at time t = t n, and l derivatives of te solution at time t = t n+1, and ten integrating te result. In practice, at least for ODEs, an appropriate extension of 4) defines iger derivatives of te j=0

7 6 Alexander Jaust, Jocen Scütz and David C. Seal solution. Eac of tese metods are of order m = l + k cf. II.12 in [22]). Because of te growing complexity of iger order derivatives see also Eqn. 3)), we rely on scemes involving only two derivatives of te unknown. Remark 2 Employed metods) In general, te two point two-derivative metod in tis work can be written in te 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 ) ), were te coefficients α i, β i, for i = 1, 2 are cosen to increase te order of accuracy or modify te region of absolute stability of te metod. In tis work, we make use of te tird-order metod wit k = 1, l = 2, given by y n+1 = y n + t 3 6) gy n ) + 2gy n+1 ) ) t2 6 ġyn+1 ), 7) and te fourt-order sceme wit k = l = 2 given by y n+1 = y n + t 2 gy n ) + gy n+1 ) ) + t2 12 ġy n ) ġy n+1 ) ). 8) Tese are te same metods tat are used to discretize te non-linear terms in [5]. Lemma 2 Stability) Te integrators 7) and 8) are tird- and fourtorder accurate, respectively, and A-stable. Te tird-order metod 7) is L- stable. Te order of accuracy is given in [22]. Here, we include a proof tat tese two metods are A, and L-stable, respectively. A plot of te stability region for te tird-order metod is presented in Fig. 1. Proof We apply eac metod to te test equation were y = λy, were λ C is a complex number. Metod 7) results in y n+1 = y n + t 3 and metod 8) becomes y n+1 = y n + t 2 λy n + 2λgy n+1 ) ) t2 6 λ2 y n+1, 9) λy n + λy n+1) + t2 12 If we define µ := λ t, ten eac metod can be written as were λ 2 y n λ 2 y n+1). 10) y n+1 = µ)y n 11) µ) = 1 + µ µ + µ2 6 12)

8 Multiderivative DG 7 for metod 7), and µ) = 1 + µ 2 + µ µ 2 + µ ) for metod 8). For metod 8), we observe tat iy) = 1 for any y R, and tat lim µ µ) = 1. Because tis function as no poles in te left alf plane C, te maximum modulus teorem indicates tat µ) < 1 for all µ C. For metod 7), we ave lim µ µ) = 0, and terefore tis metod as stiff decay. To obtain L-stability, we likewise need only sow tat iy) 1 for any y R because tis function as no poles in te left alf plane. Omitting details for brevity, it can be sown tat iy) 2 = 4y2 + 9) y 4 + 4y = 4y y 4 + 4y y y 2 = 1, 14) + 36 in wic case iy)) 2 1 for any y R. 80 Stability region 1 1 Norm along te imaginary axis Imµ) µ)) Reµ) Imµ) Fig. 1 Plots of stability regions. Here, we plot te region of absolute stability for te L- stable tird-order metod defined in 7). In te left panel we plot µ) for various values of µ C, and in te rigt panel we plot iy), were is defined in Eqn. 12), and y is a real number. Te stability region for te fourt order metod is te left alf plane C, and terefore it is left out for brevity. Remark 3 One final observation is tat te stability polynomials for te tirdand fourt-order metods are identical to te Padé approximants R 1,2 µ), R 2,2 µ), respectively for e µ. Applying 6) 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

9 8 Alexander Jaust, Jocen Scütz and David C. Seal 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 [21,48], 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 6) 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 ) ), 15) 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. We neglect tis, for ease of exposition. Te metod to be presented relies on te quantities σ := w x, τ := σ x = w xx, ψ := τ x = w 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, 16) τ, ϕ ) Ωk + σ, ϕ ) x ) Ωk σ, ϕ n Ωk = 0 ϕ V, 17) σ, ϕ ) Ωk + w, ϕ ) x ) Ωk ŵ, ϕ n Ωk = 0 ϕ V. 18) As is customary, we ave defined te abbreviations f, g) Ωk := Ω k Ω k fgdx, f, g Ωk := Ω k Ω k fgdσx).

10 Multiderivative DG 9 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 [48]. Te corresponding fluxes read ŵ = w +, σ = σ, τ = τ +, ψ = ψ, 19) were we stick to te convention tat w refers to te left value of w at te interface, and w + refers to te rigt value. We summarize tese quantities in an auxiliary variable x V 4 =: X, given by x := w, σ, τ, ϕ ). 20) Tis simplifies te defining equations 16) 18) for σ, τ and ϕ as N aux x, ϕ ) = 0 ϕ V 3. 21) For a convenient notation we use te following abbreviations V 4 := V V V V, V 3 := V V V. 22) Note tat tis does not refer to te polynomial degree tat is used. Remark 4 Lifting operators) It is pointed out in [48] tat it is possible to express te variable σ in terms of w via lifting operators, and subsequently τ and ψ in terms of w as well. Tis comes at te expense of computing lifting operators, i.e., to locally solve linear systems of equations in eac cell c.f. [2]). Wit tese preliminaries, we now consider te semi-discretization 15) see also 3)) once again. It is well-known ow to spatially discretize Rw) using te DG metod Rw ), ϕ ) Ωk N R x, ϕ ) 23) := fw ) εσ, ϕ ) x ) Ωk fw +, w ) ε σ, ϕ n 24) 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) as given in equation 3) 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 N Ω k R 2x, ϕ ) := 25) f w ) 2 σ εf ) w )τ, ϕ ) x + f ŵ Ω ) 2 σ εf ŵ ) τ, ϕ n k Ω k 26) + εdf, 2 ϕ ) x εd 2 )Ω k f, ϕ ) x ε 2 ) ψ, ϕ + ε 2 ψ, ϕ Ω Ω n k k Ω k. 27)

11 10 Alexander Jaust, Jocen Scütz and David C. Seal Again, te fluxes ŵ, σ, τ and ψ are te LDG fluxes wit alternating evaluation, see 19). D 2 f denotes an approximation to fw) xx, see Remark 5. Remark 5 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: 1. Linear equation, i.e., fw) = cw. In tis case, fw) xx = cw xx, and a suitable coice is 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 := σ 2 + wτ, D2 f := σ 2 + ŵ τ. 3. A similar procedure as wit Burgers equation is possible wit any flux function - also for Euler s 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), ϕ2) ) X wit ϕ 1) V. Furtermore, let te semi-linear form N be given by V 3 and ϕ2) N x, ϕ ) := were N eq x, ϕ 2) ) is given by ) N aux x, ϕ 1) ) N eq x, ϕ 2) ) N eq x, ϕ 2) ) := α 1N R x n, ϕ 2) ) + α 2N R x n+1, ϕ 2) ) ) + t β 1 N R 2x n, ϕ 2) ) + β 2N R 2x n+1, ϕ 2) ). Te coefficients α i and β i are te same as in Remark 2 and are cosen to modify order of accuracy or stability of te time integrator. 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. 1 t Ω k Note tat te first component wic is indeed vector-valued in R 3, i.e., 0 R 3 ) stems from 16) 18), wile te second component is te discretized version of equation 15).

12 Multiderivative DG 11 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 15), i.e., tere olds: 1 t 0 ) w, t n+1 ) w, t n ), ϕ Ω k ) N x, ϕ ) = O t q ) were q = 3 for integrator 7) and q = 4 for integrator 8). 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 Numerical results: 1D examples In tis section, we present numerical results for te newly developed sceme. In eac case, we demonstrate tat te optimal order of convergence is met. 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 ) L2 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.

13 12 Alexander Jaust, Jocen Scütz and David C. Seal Numerical results for different values of te polynomial order p of te ansatz space are sown in Fig. 2 for te tird-order integrator 7) left) and te fourtorder integrator 8) 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 Error e 10 7 Error e p = 0 p = 1 p = 2 p = p = 0 p = 1 p = 2 p = 3 Fig. 2 Numerical results for te eat equation. In bot computations, we coose te ratio t to be one. Temporal integration is performed via te tird-order accurate integrator 7) x left) and te fourt-order integrator 8) rigt). Tabulated results are given in Tab 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. 3, again for te tird-order left) and te fourtorder 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. 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. For tis example, we also ran experiments for a longer time T. For brevity, we do not report te results ere, but simply state tat te metods beave as expected: optimal orders of accuracy are met.

14 Multiderivative DG Error e 10 6 Error e p = 0 p = 1 p = 2 p = p = 0 p = 1 p = 2 p = 3 Fig. 3 Numerical results for te convection equation. Temporal integration is done via te tird-order integrator 7) left) and te fourt-order integrator 8) 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. Tabulated results are given in Tab 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. Te initial conditions for tis test problem are given by w 0 x) = sin2πx), and te ratio t x is cosen to be one for all cases. Numerical results are presented in Figs. 4 and 5. In Fig. 4, we demonstrate a convergence study for various values of te polynomial degree p. In Fig. 5, we compare errors versus time to solution against oter competing time stepping metods. We coose two implicit DIRK metods to compare against because we ave used tese metods in te past and found tat tey are efficient [29]. Moreover, because tey are implicit metods written in te same framework, we can make use of identical data structures in order to provide fair timing results. Te fourt-order metod we use can be found in Hairer and Wanner s text [24], and te tird-order metod is attributed to Cas [4]. To keep te plot from becoming too complicated, we only sow results for p = 3, owever, all oter results look similarly. One can observe tat bot te error levels and also te timing results are more or less te same. 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.

15 14 Alexander Jaust, Jocen Scütz and David C. Seal Error e 10 6 Error e p = 0 p = 1 p = 2 p = p = 0 p = 1 p = 2 p = 3 Fig. 4 Numerical results for te convection-diffusion equation. In bot computations, we coose te ratio t to be one. Temporal integration is done via te tird-order integrator x 7) left) and te fourt-order integrator 8) rigt). Tabulated results are given in Tab p = 3: Multiderivative p = 3: Cas 10 0 p = 3: Multiderivative p = 3: Hairer/Wanner Error e 10 6 Error e CPU time CPU time Fig. 5 Timing results for te convection-diffusion equation. Te settings are te same as in Fig. 4, owever tis time, we plot error versus computational time. Tird order plots left) use Cas s DIRK metod, fourt-order plots use Hairer/Wanner s DIRK metod. Te plots indicate tat bot error levels and timings are very comparable. Tabulated results are given in Tab Convection-diffusion: Discontinuous initial conditions. In tis section, we consider discontinuous initial data w 0 x) = Hsin2πx 0.3)))e 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. 20) 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, tis problem as non-smoot initial conditions, and terefore w 0 does not exist. At least not in te classical sense, as it is a Dirac measure.) For tis case, one alternative is to compute σ 0, τ 0 and ψ0 as a solution to N auxx 0, ϕ ) = 0

16 Multiderivative DG 15 2 w, 0) w, T ) w, T ) x x Fig. 6 Approximate versus exact solution at time T = 0.5 rigt) to te convection-diffusion equation wit discontinuous initial datum left). Results are computed wit p = 2, = 1 16 and te tird-order integrator 7). see also 16)-18) for te defining equations) for all ϕ V 3 for a given w = Π V w 0. Tis is te ansatz we pursue in tis section. In Fig. 6, 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. 5.4 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. 7. Te exact solution is computed using te Cole-Hopf transformation [28]. 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. Tis is appealing because we already ave access to tese iger derivatives.

17 16 Alexander Jaust, Jocen Scütz and David C. Seal Error e 10 6 Error e p = 0 p = 1 p = 2 p = p = 0 p = 1 p = 2 p = 3 Fig. 7 Numerical results for Burgers equation. In bot computations, we coose te ratio t to be one. Temporal integration is done via te tird-order integrator 7) left) and te x fourt-order integrator 8) rigt). Tabulated results are given in Tab D: 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 + 28) 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 equations 7)-8), but we discretize te resulting system using te ybridized discontinuous Galerkin [10, 36, 42] metod. For Poisson s equation, tis discretization is equivalent to a variant of) te LDG metod [10]. A preliminary investigation tat couples two-derivative Runge-Kutta metods wit HDG for a linear advection equation can be found in in [30]. On te one side, te HDG metod really depends on efficient implicit time integrators due to te stiffness of te system of equations. Terefore, it is especially important to find new implicit time integrators tat may be better tan te currently employed BDF and DIRK metods. On te oter side, 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

18 Multiderivative DG 17 additional unknowns are introduced by te spatial derivatives from te twoderivative time discretization. In tis section, we only consider convection equations, treating diffusive parts is left for future work, as te treatment of iger-order derivatives in te HDG metod is by far not standard. See [6,9] for te extension of HDG to iger-order derivatives.) 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 29) w) 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. 29). 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 [10]. 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. For te approximation of second order spatial derivatives introduced by a twoderivative 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 := σ, w, λ )

19 18 Alexander Jaust, Jocen Scütz and David C. Seal tat stems 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 given previously, 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. 30) Finally, te discretization of R and R 2 for tis first-order PDE is given by ) ) Rw ), ϕ 2) N R x, ϕ 2) ) := fw ), ϕ 2) 2), f, ϕ n Ω k and ) R 2 w ), ϕ 2) N R 2x, ϕ 2) ) Ω k ) := D fw, σ ), ϕ 2) were Ω k + Ω k Ω k D f, ϕ 2), n, Ω k f w) fw) = f w)f 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. Definition 2 HDG metod) Let ϕ = X. Furtermore, let te semilinear form N be given by N aux x, ϕ 1) ) N x, ϕ ) := N eq x, ϕ 2) ), N yb x, ϕ 3) )

20 Multiderivative DG 19 wit N aux x, ϕ 1) ) as defined in equation 30), were N eqx, ϕ 2) ) is given by N eq x, ϕ 2) ) :=α 1N R x n, ϕ 2) ) + α 2N R x n+1, ϕ 2) ) ) + t β 1 N R 2x n, ϕ 2) ) + β 2N R 2x n+1, ϕ 2) ), and te ybrid term is given by N yb x, ϕ 3) ) := n+1 α 2 f + β2 D f n+1, ϕ 3) Te brackets denote te jump operator v = v n v + n Γ. wit vx) ± being vx) ± = lim ɛ 0 vx ± ɛn), x Ω k 31) 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 = N x, ϕ ) ϕ X. = Remark 6 Number of unknowns) All evaluations on elements only depend on local values of w and σ, and terefore te total number of unknowns 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 [10]. 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 Numerical results: 2D examples 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

21 20 Alexander Jaust, Jocen Scütz and David C. Seal 7.1 Linear advection equation We first examine a system of linear advection equations. It can be written as in Eqn. 28). 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, 32) Te vector of unknowns is w = w 1, w 2 ) T. Te matrices for tis linear system are given by 1 8) A 1 = , A 2 = 5 ) ) 3 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 ) 5 0 D A1 =, D A2 = ) 1, S = ), S = ). 34) After coosing te initial conditions to be ) sinπx + y)) w 0 x) =, 35) sinπx + y)) and taking into account periodic boundary conditions, we write te exact solution as ) sinπx + y + t)) wx, t) =. 36) sinπx + y + t)) We compute solutions on meses tat are presented in Fig. 8. Results are Fig. 8 Left : Initial mes. Rigt: Mes after a total of tree refinements. presented in Fig. 9 for te 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 p = 3, te metod is still tird-order accurate, but it as a lower error tan in te case wit p = 2. Te fourt-order integrator, owever, does not acieve fourt-order in time. In te case p < 3, te metod

22 Multiderivative DG Error e 10 3 Error e p = 0 p = 1 p = 2 p = p = 0 p = 1 p = 2 p = Error e 10 3 Error e p = 0 p = 1 p = 2 p = p = 0 p = 1 p = 2 p = Fig. 9 Numerical results for te linear coupled advection equation. Temporal integration is performed wit te tird-order integrator 7) left) and te fourt-order 8) integrator rigt). We sow te results for components w 1 top) and w 2 bottom). In all computations we coose te ratio t to be to ensure stability of te numerical metod. Tabulated x results are given in Tab. 6 and 7. gets close to te expected order of p + 1 wile for p = 3 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, 37) 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 ),

23 22 Alexander Jaust, Jocen Scütz and David C. Seal Error e 10 4 Error e p = 0 p = 1 p = 2 p = p = 0 p = 1 p = 2 p = Fig. 10 Numerical results for te Euler equations. Temporal integration is done via te tird-order integrator 7) left) and te fourt-order 8) integrator rigt). We sow te error in te density ρ. In all computations we coose te ratio t to be 0.05 to ensure x stability of te metods. Tabulated results are given in Tab. 8. 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 [31] 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 = 1. 38) A convergence study is presented in Fig. 10, 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 p = 2 to p = 3 decreased te error level, wereas te slope of te error grap stayed almost constant cf. Fig. 9). 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 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. For diffusion dominated problems, te integrators yield a stable sceme and te runtimes of te multiderivative scemes are reasonable compared to classical DIRK metods witout te code being

24 Multiderivative DG 23 optimized for efficiency. However, despite te fact tat we observe tat te time integrators are not uniformly stable for convection dominated problems, tese new metods work well for diffusion dominated problems and ave reasonable runtimes. 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. Furtermore, te extension of tis metodology to te full Navier-Stokes equations - also for HDG - is of extreme importance and te subject of future work. Acknowledgements. We would like to tank te anonymous reviewers for teir elpful comments and suggestions to improve te quality of tis work. A Tabulated results In tis section we present complete error tables for te data presented summarily trougout te paper. p = 0 order p = 1 order p = 2 order p = 3 order 5.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e p = 0 order p = 1 order p = 2 order p = 3 order 5.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Table 1 Numerical results for te eat equation. Tabulated results of tird-order top) and fort-order bottom) integrator as displayed in Fig. 2. References 1. Alexander, R.: Diagonally implicit Runge-Kutta metods for stiff O.D.E. s. SIAM Journal of Numerical Analysis 14, ) 2. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin metods for elliptic problems. SIAM Journal of Numerical Analysis 39, )

25 24 Alexander Jaust, Jocen Scütz and David C. Seal p = 0 order p = 1 order p = 2 order p = 3 order 5.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e p = 0 order p = 1 order p = 2 order p = 3 order 5.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Table 2 Numerical results for te convection equation. Tabulated results of tird-order top) and fourt-order bottom) integrator as displayed in Fig. 3. p = 0 order p = 1 order p = 2 order p = 3 order 5.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e p = 0 order p = 1 order p = 2 order p = 3 order 5.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Table 3 Numerical results for te convection-diffusion equation wit smoot initial condition. Tabulated results of tird-order top) and fourt-order bottom) integrator as displayed in Fig Banks, J., Hensaw, W.: Upwind scemes for te wave equation in second-order form. Journal of Computational Pysics 231, ) 4. Cas, J.: Diagonally implicit Runge-Kutta formulae wit error estimates. Journal of te Institute of Matematics and its Applications 24, ) 5. Causley, M.F., Co, H., Cristlieb, A.J., Seal, D.: Metod of lines transpose: Hig order L-stable ON) scemes for parabolic equations using successive convolution. arxiv

26 Multiderivative DG 25 error order time[s] 5.000e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+2 error order time[s] 5.000e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+2 error order time[s] 5.000e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+2 error order time[s] 5.000e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+2 Table 4 Results for te convection-diffusion equation. Tables wit errors and computing times used in Fig. 5. In clockwise direction starting at te top left te results of te tird order integrator, te DIRK metod of Cas, te DIRK metod of Hairer and Wanner and te fourt order integrator are sown. p = 0 order p = 1 order p = 2 order p = 3 order 5.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e p = 0 order p = 1 order p = 2 order p = 3 order 5.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Table 5 Numerical results for Burgers equation. Tabulated results of tird-order top) and fourt-order bottom) integrator as displayed in Fig. 7. preprint arxiv: ) 6. Cen, Y., Cockburn, B., Dong, B.: Superconvergent HDG metods for linear, stationary, tird-order equations in one-space dimension. Matematics of Computation 2015) 7. Cristlieb, A.J., Feng, X., Seal, D.C., Tang, Q.: A ig-order positivity-preserving singlestage single-step metod for te ideal magnetoydrodynamic equations. arxiv preprint arxiv: )