hal , version 1-4 May 2009

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1 EXPLICIT RUNGE KUTTA SCHEMES AND FINITE ELEMENTS WITH SYMMETRIC STABILIZATION FOR FIRST-ORDER LINEAR PDE SYSTEMS ERIK BURMAN, ALEXANDRE ERN, AND MIGUEL A. FERNÁNDEZ Abstract. We analyze explicit Runge Kutta scemes in time combined wit stabilized finite elements in space to approximate evolution problems wit a first-order linear differential operator in space of Friedrics-type. For te time discretization, we consider explicit second- and tird-order Runge Kutta scemes. We identify a general set of properties on te spatial stabilization, encompassing continuous and discontinuous finite elements, under wic we prove stability estimates using energy arguments. Ten, we establis L 2 -norm error estimates wit (quasi-)optimal convergence rates for smoot solutions in space and time. Tese results old under te usual CFL condition for tird-order Runge Kutta scemes and any polynomial degree in space and for second-order Runge Kutta scemes and first-order polynomials in space. For second-order Runge Kutta scemes and iger polynomial degrees in space, a tigtened 4/3-CFL condition is required. Numerical results are presented for te advection and wave equations. Key words. First-order PDEs, transient problems, stabilized finite elements, explicit Runge Kutta scemes, stability, convergence. al , version 1-4 May 2009 AMS subject classifications. 65M12, 65M15, 65M60, 65L06 1. Introduction. Let Ω be an open, bounded, Lipscitz domain in R d and let T > 0 be a finite simulation time. We consider te following linear evolution problem t u + Au = f in Ω (0,T), (1.1) completed wit suitable initial and boundary conditions specified below. Here, u : Ω (0,T) R m, m 1, is te unknown and f : Ω (0,T) R m is te source term. Moreover, A is a first-order linear differential operator in space endowed wit a symmetry property specified below (te operator A can also accommodate a zero-order term). Typical examples include advection problems and linear wave propagation problems in electromagnetics and acoustics. Our goal is to analyze approximations to (1.1) using explicit Runge Kutta (RK) scemes in time and finite elements wit symmetric stabilization in space. Explicit RK scemes are popular metods to approximate in time systems of ordinary differential equations. In te context of space discretization by discontinuous Galerkin (DG) metods, explicit RK scemes ave been developed by Cockburn, Su, and co-workers [11, 10, 8] and applied to a wide range of engineering problems (see, e.g., [9] and references terein). Tis is in stark contrast wit te case of space discretization by continuous finite elements were, to our knowledge, stabilization tecniques ave not yet been analyzed in combination wit explicit RK scemes. In particular, SUPG-type stabilization seems not to be compatible wit explicit RK scemes. In fact, te only viable explicit metod wit continuous approximation in space for te present evolution problems is, to our knowledge, te metod of caracteristics [15, 25]. Alternatively, implicit metods can be considered (see, e.g., [19, 16, 7]), i.e. Department of Matematics, Mantell Building, University of Sussex, Brigton, BN1 9RF United Kingdom (E.N.Burman@sussex.ac.uk) Université Paris-Est, CERMICS, Ecole des Ponts, Marne la Vallée Cedex 2, France (ern@cermics.enpc.fr) INRIA, CRI Paris-Rocquencourt, Rocquencourt BP 105, F Le Cesnay Cedex, France (miguel.fernandez@inria.fr) 1

2 2 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ based on A-stable time discretizations, or semi-implicit metods [7], resulting from te combination of an A-stable sceme wit an appropriate explicit treatment of te stabilization operator. Te starting point of our analysis is te observation tat owing to te properties of te differential operator A, tere is a real number λ 0 s.t. for te exact solution, (Au,u) L 1 2 (Mu,u) L, Ω λ 0 u 2 L, (1.2) al , version 1-4 May 2009 were M is a non-negative R m,m -valued boundary field depending on te boundary conditions. Here, we ave set L := [L 2 (Ω)] m, (, ) L denotes te usual scalar product in L wit associated norm L, and (, ) L, Ω denotes te usual scalar product in [L 2 ( Ω)] m. As a result, an energy can be associated wit te evolution problem (1.1). Indeed, taking te L-scalar product of (1.1) by u, exploiting (1.2), and integrating in time, it is inferred using Gronwall s lemma tat solutions to (1.1) satisfy te energy estimate max t [0,T] u 2 L + T 0 (Mu,u) L, Ω dt C, (1.3) were te constant C depends on te initial condition, te source f, te simulation time T, and te parameter λ 0. Tis implies in particular tat te energy, defined as te L-norm of te solution, is controlled at any time. Following te seminal work of Levy and Tadmor [24], our analysis of explicit RK scemes wit stabilized finite elements inges on energy estimates. Te crucial point is tat explicit RK scemes are anti-dissipative (tat is, tey produce energy at eac time step), and tis energy production needs to be compensated by te dissipation of te stabilization sceme in space. In [24], a so-called coercivity condition was proposed on te discrete differential operator in space, and wit tis condition, te stability of te usual RK3 and RK4 scemes was proven under a CFL-type condition. Te coercivity condition in [24] can for instance be satisfied if an artificial viscosity is used for space stabilization. However, artificial viscosity yields suboptimal convergence estimates in space as soon as finite elements wit polynomials of degree 1 are used. In te present paper, we improve on tis point by establising stability estimates for a wide class of ig-order finite element metods wit symmetric stabilization. Hig-order finite element metods do not satisfy te above coercivity condition. Instead, we derive ere a sarper set of conditions on te stabilization and proceed along a different pat tan in [24] for te stability analysis, still relying on energy arguments. Furtermore, we additionally derive energy error estimates tat are optimal in time and quasi-optimal in space provided te exact solution is smoot enoug. We also consider fully unstructured simplicial meses. A salient feature is tat our conditions allow for a unified analysis of several ig-order stabilized finite element metods encountered in te literature. Examples include in te context of continuous finite elements, e.g., interior penalty of gradient jumps [2, 3], local projection [1, 26], subgrid viscosity [18, 19], or ortogonal subscale stabilization [12, 13], and also include discontinuous finite elements (DG metods) [23, 22, 17, 14]. Incidentally, a noteworty point is tat DG metods can be cast into te same unified framework as stabilized finite element metods, indicating tat all tese metods essentially sare te same stability properties. Explicit RK scemes come in various forms; see, e.g., [20]. Here, we present results for two-stage second-order and tree-stage tird-order scemes (abbreviated as RK2 and RK3, respectively). Tese scemes are written in a specific form suitable

3 RK-FEM WITH SYMMETRIC STABILIZATION 3 al , version 1-4 May 2009 for te present analysis, and we verify tat usual RK2 and RK3 scemes encountered in te literature can be cast into tis form. Our main results can be summarized as follows: under te usual CFL condition τ (/σ) were τ is te time step, te minimal mes size, σ a reference velocity, and a dimensionless constant, an energy error estimate of te form O(τ 2 + 3/2 ) for te RK2 sceme and piecewise affine finite elements; under te tigtened 4/3-CFL condition τ (/σ) 4/3, an energy error estimate of te form O(τ 2 + p+1/2 ) for te RK2 sceme and finite elements wit polynomials of total degree p wit any p 2; under te usual CFL condition τ (/σ), an energy error estimate of te form O(τ 3 + p+1/2 ) for te RK3 sceme and finite elements wit polynomials of total degree p wit any p 1. To te best of our knowledge, te above results are new for continuous finite element metods. As suc, tey provide an attractive alternative to te metod of caracteristics since te present metods are more easily extendible to iger order. For DG metods, te two above results for RK2 scemes ave been obtained by Zang and Su in te more general context of nonlinear scalar conservation laws [27] and symmetrizable systems of nonlinear conservation laws [28]. Our result for te RK3 sceme is, to te best of our knowledge, new. Moreover, te present proofs for RK2 scemes on linear PDE systems allow to identify more directly te stability properties of DG metods tat play a role in te analysis. Tis paper is organized as follows. In 2, we present te continuous and discrete settings and state te conditions on te stabilization of te finite element metod allowing for te unified analysis. In 3 and 4, we treat RK2 and RK3 scemes, respectively. Numerical results illustrating te teory are presented in 5. Finally, some conclusions and lines for future work are drawn in Te setting. Tis section presents te continuous and discrete settings togeter wit some examples. We also state te conditions on te stabilization of te finite element metod allowing for te unified analysis Te continuous problem. Let {A i } 1 i d be fields in [L (Ω)] m,m s.t. A i is symmetric a.e. in Ω, i {1,...,d}, (2.1) d Λ := i A i [L (Ω)] m,m. (2.2) i=1 Te differential operator A in (1.1) is A := d A i i. (2.3) i=1 For furter use, we set σ := max 1 i d A i [L (Ω)] m,m. Assuming tat te PDE system (1.1) is written in non-dimensional form for u, te components of te fields A i scale as velocities, and te quantity σ represents a maximum wave speed. Let n = (n 1,...,n d ) denote te outward unit normal to Ω. Define te boundary matrix field D [L ( Ω)] m,m s.t. a.e. on Ω, D := d A i n i, (2.4) i=1

4 4 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ and observe tat D takes, by construction, symmetric values. Te boundary condition on (1.1) enforces at all times t (0,T), (D M)u Ω = 0, (2.5) were te boundary matrix field M [L ( Ω)] m,m is non-negative a.e. on Ω, and M [L ( Ω)] m,m C Mσ for some constant C M. Te initial condition is u(,0) = u 0 in Ω wit u 0 L, and te source term is suc tat f C 0 (0,T;L). Wen deriving convergence rates, we assume tat u C 0 (0,T;[H p+1 (Ω)] m ) if finite elements of degree p are used, and tat u C l (0,T;L) and f C l 1 (0,T;L) wit l = 3 for RK2 and l = 4 for RK3. Since M is non-negative, te seminorm is well-defined. Define te bilinear form v M := (Mv,v) 1/2 L, Ω, (2.6) a(v,w) = (Av,w) L ((M D)v,w) L, Ω. (2.7) al , version 1-4 May 2009 A crucial consequence of properties (2.1) and (2.2) is tat integration by parts yields Letting λ 0 := 1 2 Λ [L (Ω)] m,m leads to a(v,v) = 1 2 v 2 M 1 2 (Λv,v) L. (2.8) a(v,v) 1 2 v 2 M λ 0 v 2 L. (2.9) We now give tree examples of evolution problems fitting te present framework. Advection: let β [L (Ω)] d wit β L (Ω) and consider te PDE Set m = 1 and t u + β u = f. (2.10) A i = β i, i {1,...,d}, (2.11) yielding D = β n. An admissible boundary condition consists in taking M = β n wic enforces u to zero on te inflow boundary. Maxwell s equations: let µ,ǫ be positive constants, set c 0 = (µǫ) 1/2, and consider te PDE system { µ t H + E = f 1, (2.12) ǫ t E H = f 2, were H is te magnetic field and E te electric field. Set m = 6, u = (µ 1/2 H,ǫ 1/2 E), and let [ ] 03,3 R i A i = c 0 Ri t, i {1,...,d}, (2.13) 0 3,3 were 0 3,3 is te null matrix in R 3,3 and (R i ) jk = ǫ jik for i,j,k {1,2,3}, ǫ jik being te Levi Civita permutation tensor. An admissible boundary condition

5 RK-FEM WITH SYMMETRIC STABILIZATION 5 is for instance to enforce a Diriclet condition on te tangential component of te electric field. Ten, D and M are given by [ ] [ ] 03,3 N 03,3 N D = c 0 N t, M = c 0 0 3,3 N t, (2.14) 0 3,3 were N = 3 i=1 n ir i R 3,3 is suc tat Nz = n z for all z R 3. Acoustics equations: let c 0 be a positive constant, and consider te PDE system { c 2 0 tp + q = f 1, (2.15) t q + p = f 2, al , version 1-4 May 2009 were p is te pressure and q te momentum per unit volume. Set m = 4, u = (c 1 0 p,q), and let [ ] 0 e t i A i = c 0, i {1,...,d}, (2.16) e i 0 3,3 were (e 1,e 2,e 3 ) denotes te Cartesian basis of R 3. An admissible boundary condition is for instance to enforce a Diriclet condition on te normal component of te flux. Ten, D and M are given by [ ] [ ] 0 n D = c t 0 n 0, M = c t n 0 0. (2.17) 3,3 n 0 3, Space discretization. Let {T } >0 be a family of simplicial meses of Ω were denotes te maximum diameter of elements in T. For simplicity, we assume tat te meses are affine. Mes faces are collected in te set F wic is split into te set of interior faces, F int, and boundary faces, Fext. For T T and for F F, L,T and L,F respectively denote te [L 2 (T)] m - and [L 2 (F)] m -norms; moreover, we define 2 L,F := F F 2 L,F. We assume tat meses are kept fixed in time and also tat te family {T } >0 is quasi-uniform; see Remark 2.1 below. Let V be a finite element space consisting of eiter continuous or discontinuous piecewise polynomials of total degree p wit p 1 (te case p = 0 is also possible for DG metods). Let π denote te L-ortogonal projection onto V. Set V () := [H p+1 (Ω)] m + V. We consider a discrete version of te bilinear form a, namely a, togeter wit a stabilization bilinear form s. Bot forms are defined on V () V. We define te linear operators A : V () V and S : V () V s.t. (v,w ) V () V, (A v,w ) L := a (v,w ), (S v,w ) L := s (v,w ). (2.18) We also define L : V () V s.t. L = A + S. (2.19) Te seminorm M defined above can be extended to V (). We now state te key design conditions on te bilinear forms a and s. Te first tree assumptions are te following:

6 6 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ (A1) for all v V, a (v,v ) = 1 2 v 2 M 1 2 (Λv,v ) L ; (A2) s is symmetric and non-negative on V () V ; (A3) te strong solution u satisfies for all t (0,T), L u = π (f t u). Assumption (A3) is a (strong) consistency property; it is equivalent to te fact tat for te strong solution u, for all t (0,T), and for all v V, a (u,v ) + s (u,v ) = a(u,v ). (2.20) Tis consistency property can be weakened for te stabilization bilinear form s. Tis is, in particular, useful wen analyzing local projection or ortogonal subscale stabilization. Te present strong consistency assumption is sufficient to analyze interior penalty stabilization and DG metods; see Section 2.4 for examples. Furtermore, owing to Assumption (A2), we can define on V () te seminorm v S := { s (v,v) v 2 M} 1/2. (2.21) Te oter assumptions concern te stability of te discrete operators S and L, namely (A4) tere is C S s.t. for all v V, al , version 1-4 May 2009 and tere is C S s.t. for all v [Hp+1 (Ω)] m, (A5) tere is C L s.t. for all z V (), v S C 1/2 S σ1/2 1/2 v L, (2.22) v π v S C S p+1/2 v [H p+1 (Ω)] m; (2.23) L z L C L (σ z L d + σ 1/2 1/2 z S ), (2.24) were denotes te broken gradient of z and L d te usual norm in L d (te broken gradient is needed wen working wit DG metods; it coincides wit te usual gradient for continuous finite elements); (A6) tere is C π s.t. for all (z,v ) V () V, (L (z π z),v ) L C π σ 1/2 z π z ( v S + v L ), (2.25) wit te norm for y V (), y := 1/2 y L d + 1/2 y L + y L,F + σ 1/2 y S. (2.26) Finally, in te piecewise affine case, tat is, p = 1 in te definition of te discrete space V, we also assume tat (A7) tere is C π s.t. for all (v,w ) V V wit p = 1, (L v,w π 0 w ) L C πσ 1/2 1/2 ( v S + v L ) w π 0 w L, (2.27) were π 0 denotes te L-ortogonal projection onto piecewise constant functions. Our analysis inges on Assumptions (A1) (A7). For furter use, we point out some useful facts associated wit tese assumptions. An important consequence of (A1) (A2) and te definition (2.21) of te seminorm S is te following dissipativity property of te discrete setting: For all v V, (L v,v ) L = v 2 S 1 2 (Λv,v ) L. (2.28)

7 RK-FEM WITH SYMMETRIC STABILIZATION 7 al , version 1-4 May 2009 Moreover, it is clear tat (A5) implies for all z V (), L z L C L σ 1/2 z. (2.29) Using inverse and trace inequalities, it is readily inferred from te definition (2.26) of te norm tat tere is C suc tat for all v V, Hence, letting C L := C L C, tere olds for all v V, v C 1/2 v L. (2.30) L v L C L σ 1 v L. (2.31) Finally, using (2.23) and usual approximation properties in finite element spaces, it is inferred tat tere is C s.t. for all v [H p+1 (Ω)] m, v π v C p+1/2 v [H p+1 (Ω)] m. (2.32) 2.3. CFL conditions. Let τ be te time step, taken to be constant for simplicity and suc tat T = Nτ were N is an integer. For 0 n N, a superscript n indicates te value of a function at te discrete time nτ, and for 0 n N 1, we set I n = [nτ,(n + 1)τ]. We assume witout loss of generality tat τ 1. We also assume tat te following, so-called usual, CFL condition olds τ (/σ), (2.33) for some positive real number. Te value of will be specified below wenever relevant. Furtermore, in te case of RK2 scemes wit polynomials of total degree 2, we will also need te so-called strengtened 4/3-CFL condition τ (/σ) 4/3, (2.34) for some positive real number. Again, te value of will be specified below wenever relevant. Since τ 1, te strengtened 4/3-CFL condition (2.34) implies te CFL condition (2.33) wit = ( ) 3/4. Remark 2.1. It is also possible to work wit sape-regular mes families. In tis case, as usual, te space scale in te CFL condition is no longer, but te smallest element diameter in te mes. Te same space scale is used in te negative powers of in Assumptions (A4) (A7) Examples. In tis section, we present two examples of discrete bilinear forms satisfying Assumptions (A1) (A7). For F F int, tere are T,T + in T suc tat F = T T +, n F is te unit normal to F pointing from T to T +, and for a smoot enoug function v tat is possibly double-valued at F, we define its jump and mean value at F as [[v]] := v T v T + and {v } = 1 2 (v T + v T +), respectively. For vector-valued functions, te jump and averages are defined componentwise as above. Te arbitrariness in te sign of [[v]] is irrelevant. Meses can possess anging nodes wen working wit discontinuous finite elements. An example wit continuous finite elements consists in considering symmetric stabilization based on inter-element jumps of te gradient of te discrete solution [2, 3, 5]. In tis case, a cip 1 (v,w) := (Av,w) L,T + 2 ((M D)v,w) L,F, (2.35) T T F F ext s cip (v,w) := (SF ext v,w) L,F + 2 F(SF int n F [[ v]],n F [[ w]]) L,F, (2.36) F F ext F F int

8 8 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ were F denotes te diameter of F. An example wit discontinuous finite elements consists in taking [17] a dg (v,w) := acip (v,w) s dg (v,w) := (S ext F F ext F F int (D F [[v]], {w }) L,F, (2.37) F v,w) L,F + F F int (S int F [[v]],[[w]]) L,F, (2.38) were D F := d i=1 A in F,i for all F F int. Te R m,m -valued fields SF ext and Sint F, wic are defined on boundary and interior faces, respectively, must satisfy te following design conditions: for te exact solution u and for all t [0,T], S ext F u = 0 on Ω, (2.39) S ext F and S int F are symmetric and non-negative, (2.40) F F ext, SF ext α 1 σi m and F F int, α 2 D F SF int α 3 σi m, (2.41) and for all F F ext and for all (y,z) [L 2 (F)] m [L 2 (F)] m, al , version 1-4 May 2009 ((M D)y,z) L,F α 4 σ 1/2 y S,F z L,F, (2.42) ((M + D)y,z) L,F α 5 σ 1/2 y L,F z S,F. (2.43) Here, α 1,...,α 5 are positive parameters, inequalities in (2.41) are meant on te associated quadratic forms, I m denotes te identity matrix in R m,m, and for F F ext, v S,F := {(SF extv,v) L,F + (Mv,v) L,F } 1/2 wic is well-defined since SF ext is nonnegative. Te absolute value D F is also well-defined since D F is, by construction, symmetric. Lemma 2.1. Assume tat te design conditions (2.39) (2.43) old and tat for all T T and for all i {1,...,d}, A i T [C 0,1/2 (T)] m,m. Ten, Assumptions (A1) (A7) old for a cip and s cip defined by (2.35) (2.36) and for a dg and s dg defined by (2.37) (2.38). Proof. Assumptions (A1) (A6) can be proven as in [5, 6] for continuous finite elements wit interior penalty and as in [17] for discontinuous finite elements. To prove Assumption (A7) for continuous finite elements, let (v,w ) V V wit p = 1 and set y = w π 0w. Since y may not be in V, we obtain (L v,y ) L = T T (Av,π y ) L,T + F F ext 1 2 ((M D)v,π y ) L,F + s cip (v,π y ). For te tird term, using (A2) and (A4) we obtain s cip (v,π y ) v S π y S v S C 1/2 S σ1/2 1/2 π y L C 1/2 S σ1/2 1/2 v S y L. Te second term is bounded similarly using (2.42) and a trace inequality to bound π y L,F. For te first term, observe tat T T (Av,π y ) L,T = T T (Av,π y y ) L,T + T T (Av,y ) L,T := T 1 + T 2.

9 RK-FEM WITH SYMMETRIC STABILIZATION 9 Following [5] using a so-called Oswald interpolate, it can be proven tat T 1 Cσ 1/2 1/2 ( v S + v L ) y L. Furtermore, since p = 1, letting A := d i=1 (π0 A i) i, T 2 = ((A A)v,y ) L,T Cσ 1/2 1/2 v L y L, T T al , version 1-4 May 2009 using te local regularity of te fields A i and an inverse inequality. Tis completes te proof of Assumption (A7) for continuous finite elements. For discontinuous finite elements, te proof is similar, but simpler since π y = y and so T 1 = 0. Te above setting can be applied to te PDE systems presented in 2.1; see [5, 17]. Advection: take SF ext γ > 0 (γ = 1 2 Maxwell s equations: take = 0 and SF int = γ β n F wit user-defined parameter amounts to so-called upwinding in te context of DG metods). [ ] 03,3 0 3,3 SF ext = c 0 0 3,3 γ 1 N t, S int N F = c 0 γ 2NF t N F 0 3,3 0 3,3 γ 3 NF t N F, (2.44) were γ 1, γ 2, and γ 3 are positive user-defined parameters and were N F is defined as N by using n F instead of n. Te operator SF int amounts to penalizing on eac interface te jump of te normal derivative (for CIP) or of te value (for DG) of te tangential components of te electric and magnetic fields. Acoustics: take [ ] [ ] SF ext 0 0 t 3 = c 0, S int γ 2 0 t 3 F = c 0, (2.45) 0 3 γ 1 n n 0 3 γ 3 n F n F were 0 3 is te null vector in R 3. Te operator S int F amounts to penalizing on eac interface te jump of te normal derivative (for CIP) or of te value (for DG) of te pressure and tat of te normal component of te momentum per unit volume. 3. Analysis for explicit RK2 scemes. Tis section is devoted to te convergence analysis of explicit two-stage RK2 scemes. First, we present te specific form of te scemes on wic we will work and sow tat usual implementations of two-stage RK2 scemes fit tis form. Ten, we derive te error equation and establis te key energy identity. Finally, we infer (quasi-)optimally convergent error upper bounds under te CFL condition (2.33) for piecewise affine finite elements and under te strengtened 4/3-CFL condition (2.34) for polynomials wit total degree 2. We will keep track of te constants to derive te CFL conditions, but not to state te error estimates. Hencefort, C denotes a generic constant, independent of te mes size and te time step, but tat can depend on f, u, te fields A i and M, te constants in Assumptions (A4) (A7), and te constants and in te CFL condition, and wose value can cange at eac occurrence. Te inequality a Cb, for positive real numbers a and b, is often abbreviated as a b. Tis convention is kept for te rest of tis work. We also set f := π f. Finally, recall tat we assume ere u C 3 (0,T;L) and f C 2 (0,T;L).

10 10 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ 3.1. Two-stage RK2 scemes. We consider scemes of te form wit te assumption tat w n = u n τl u n + τf n, (3.1) u n+1 = 1 2 (un + w n ) 1 2 τl w n τψn, (3.2) ψ n = f n + τ t f n + δ n, δ n L τ 2. (3.3) Tere are many ways of writing explicit two-stage RK2 scemes. Since te space differential operator is linear, tey all amount in te omogeneous case (f = 0) to u n+1 = u n τl u n τ2 L 2 u n. (3.4) Two examples of two-stage RK2 scemes tat fit te present form are: Te second-order Heun metod wic is usually written in te form (3.1) (3.2) wit ψ n = f n+1. (3.5) al , version 1-4 May 2009 Assumption (3.3) obviously olds. Te second-order Runge metod (also called te improved forward Euler metod) wic is usually written in te form k 1 = L u n + f n, (3.6) k 2 = L (u n τk 1) + f n+1/2, (3.7) u n+1 = u n + τk 2. (3.8) It is readily verified tat (3.6) (3.8) can be rewritten in te form (3.1) (3.2) wit Assumption (3.3) obviously olds. Define ψ n = 2f n+1/2 f n. (3.9) ξ n = u n π u n, ζ n = w n π w n, (3.10) ξ n π = u n π u n, ζ n π = w n π w n, (3.11) wit w = u + τ t u. Using tese quantities, te errors can be written as u n u n = ξ n π ξ n, w n w n = ζ n π ζ n. (3.12) Te convergence analysis proceeds as follows. Since upper bounds on ξπ n and ζπ n readily result from standard approximation properties in finite element spaces, we observe tat error upper bounds can be derived by obtaining upper bounds on ξ n and ζ n in terms of ξn π and ζπ n and ten using te triangle inequality. To tis purpose, we first identify te error equation governing te time evolution of ξ n and ζn. Te form of tis equation is similar to (3.1) (3.2) wit data depending on ξπ, n ζπ, n f, and u. Ten, we establis an energy identity associated wit (3.1) (3.2), wence te desired upper bounds on ξ n and ζn are inferred.

11 RK-FEM WITH SYMMETRIC STABILIZATION Te error equation. Our first lemma identifies te equations governing te quantities ξ n and ζn. Lemma 3.1. Tere olds wit ζ n = ξ n τl ξ n + τα n, (3.13) ξ n+1 = 1 2 (ξn + ζ n ) 1 2 τl ζ n τβn, (3.14) α n = L ξ n π, β n = L ζ n π π η n + δ n, (3.15) wit η n = τ 1 I n (t n+1 t) 2 ttt udt. Proof. Recalling Assumption (A3), namely π t u n = L u n + f n, yields π w n = π u n τl u n + τf n. Subtracting tis equation from (3.1) yields (3.13). To derive (3.14), observe tat u n+1 = u n + τ t u n τ2 tt u n τηn, al , version 1-4 May 2009 and projecting yields Moreover, wence π u n+1 = π w n τ2 π tt u n τπ η n = 1 2 (π u n + π w n ) 1 2 τl u n τfn τ2 π tt u n τπ η n. τπ tt u n = τ t (π t u n ) = τl t u n + τ t f n = L (w n u n ) + τ t f n, π u n+1 = 1 2 (π u n + π w n ) 1 2 τl w n τ(π η n + f n + τ t f n ). Subtracting tis equation from (3.2) yields (3.14). For furter use, it is convenient to observe tat (3.13) (3.14) imply ξ n+1 = ζ n 1 2 τl (ζ n ξ n ) τ(βn α n ). (3.16) 3.3. Energy identity and stability. Our next step is to derive an energy identity, ten leading to our main stability estimate. Lemma 3.2 (Energy identity). Tere olds ξ n+1 2 L ξ n 2 L + τ ξ n 2 S + τ ζ n 2 S = ξ n+1 ζ n 2 L + τ(α,ξ n ) n L + τ(β,ζ n ) n L τ(λξn,ξ) n L τ(λζn,ζ) n L. (3.17) Proof. Multiply (3.13) by ξ n and (3.14) by 2ζn, sum bot equations, and use (2.28) to infer (L ξ n,ξn ) L = ξ n 2 S 1 2 (Λξn,ξn ) L and (L ζ n,ζn ) L = ζ n 2 S 1 2 (Λζn,ζn ) L. Remark 3.1. Te quantity ξ n+1 ζ n 2 L appearing in te rigt-and side of te energy identity (3.17) is te anti-dissipative term associated wit te explicit nature of te RK2 sceme. Tis term essentially amounts to a second-order derivative in time.

12 12 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ Lemma 3.3 (Stability). Under te usual CFL condition (2.33) for any positive real number, tere olds ξ n+1 2 L ξ n 2 L τ ξn 2 S τ ζn 2 S ξ n+1 ζ n 2 L + Cτ(τ 4 + ξπ n 2 + ζπ n 2 + ξ n 2 L). (3.18) Proof. Starting wit te energy identity (3.17), we bound te last four terms in te rigt-and side. (i) We first bound α n and βn. Let us prove tat and tat for all v V, τ α n L τ 1/2 ξ n π, τ β n L τ 1/2 ζ n π + τ 3, (3.19) τ(α n,v ) L τ ξ n π ( v S + v L ), (3.20) τ(β n,v ) L τ ζ n π ( v S + v L ) + τ 3 v L. (3.21) al , version 1-4 May 2009 Using te definition of α n, te bound (2.29), and te CFL condition (2.33) to eliminate te factor 1/2 yields τ α n L = τ L ξ n π L τc L σ 1/2 ξ n π τ 1/2 ξ n π. Similarly, using te definition of β n, te assumption on δn, and te regularity of te strong solution u yields τ β n L = τ L ζ n π π η n + δ n L τ L ζ n π L + τ 3 τ 1/2 ζ n π + τ 3. Tis proves (3.19). In addition, owing to Assumption (A6), τ(α n,v ) L = τ(l ξ n π,v ) L τ ξ n π ( v S + v L ), and similarly, using a Caucy Scwarz inequality, τ(β n,v ) L = τ(l ζ n π,v ) L +τ( π η n +δ n,v ) L τ ζ n π ( v S + v L )+τ 3 v L. (ii) Owing to te bounds (3.20) and (3.21), τ(α n,ξ n ) L +τ(β n,ζ n ) L τ ξ n π ( ξ n S + ξ n L )+τ ζ n π ( ζ n S + ζ n L )+τ 3 ζ n L. Moreover, it is inferred from (3.13) using te triangle inequality, te bound (2.31), and te CFL condition (2.33) tat Hence, owing to (3.19), ζ n L ξ n L + τ L ξ n L + τ α n L ξ n L + τ α n L. ζ n L ξ n L + τ 1/2 ξ n π ξ n L + ξ n π, since τ 1. Collecting tese bounds and using Young inequalities yields τ(α n,ξ n ) L + τ(β n,ζ n ) L 1 2 τ ξn 2 S τ ζn 2 S + Cτ(τ 4 + ξ n π 2 + ζ n π 2 + ξ n 2 L). (iii) Finally, 1 2 τ(λξn,ξ n ) L τ(λζn,ζ n ) L τ ξ n 2 L + τ ζ n 2 L τ ξ n 2 L + τ ξ n π 2, since ζ n L ξ n L + ξ n π. Tis concludes te proof.

13 RK-FEM WITH SYMMETRIC STABILIZATION Error estimates. Starting from te stability estimate (3.18), tere are two ways to bound te positive term ξ n+1 ζ n 2 L appearing in te rigt-and side. In te general case p 2, te strengtened 4/3-CFL condition (2.34) is needed. By proceeding differently and using Assumption (A7) for p = 1, it is possible to control tis term using only te usual CFL condition (2.33) General case: 4/3-CFL condition. Te next teorem provides a general a priori error estimate in te general case p 2 under te strengtened 4/3-CFL condition. Teorem 3.1. Assume tat u C 3 (0,T;L) C 0 (0,T;[H p+1 (Ω)] m ). Under te strengtened 4/3-CFL condition (2.34) for any positive real number, tere olds u N u N L + ( N 1 n=0 1 2 τ un 2 S τ wn 2 S) 1/2 τ 2 + p+1/2. (3.22) al , version 1-4 May 2009 Proof. Te proof is decomposed into tree steps. (i) Bound on ξ n+1 ζ n 2 L. Starting from (3.16) and using (3.13) leads to ξ n+1 ζ n = 1 2 τl (ζ n ξ n ) τ(βn α n ) = 1 2 τ2 L 2 ξ n τ(βn α n τl α n ). Set R n = 1 2 τ(βn αn τl α n ). Using te triangle inequality, te bound (2.31), and te CFL condition (2.33) yields so tat (3.19) implies As a result, ξ n+1 R n L τ β n L + τ α n L, R n L τ 3 + τ 1/2 ζ n π + τ 1/2 ξ n π. ζ n 2 L τ 4 L 2 ξ n 2 L + τ(τ 5 + ξ n π 2 + ζ n π 2 ). (ii) Using te above bound togeter wit te stability estimate (3.18) leads to ξ n+1 2 L ξ n 2 L τ ξn 2 S τ ζn 2 S τ 4 L 2 ξ n 2 L +τ(τ 4 + ξπ n 2 + ζπ n 2 + ξ n 2 L), since τ 5 τ 4. Te strengtened 4/3-CFL condition togeter wit (2.31) imply Hence, τ 4 L 2 ξ n 2 L τ ξ n 2 L. ξ n+1 2 L ξ n 2 L τ ξn 2 S τ ζn 2 S τ ξ n 2 L + τ(τ 4 + ξπ n 2 + ζπ n 2 ). (iii) Summing over n in te above estimate and using Gronwall s lemma, it is inferred tat N 1 N 1 ξ N 2 1 L + 2 τ( ξn 2 S + ζ n 2 S) τ 4 + τ( ξπ n 2 + ζπ n 2 ), n=0 n=0

14 14 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ and using te approximation property (2.32) yields N 1 ξ N 2 1 L + 2 τ( ξn 2 S + ζ n 2 S) τ 4 + 2p+1, n=0 wence (3.22) readily follows using a triangle inequality and te fact tat u u n S = u n S and w w n S = w n S. Remark 3.2. Altoug tere is no specific limit to te value of te constant in te 4/3-CFL condition for te convergence result of Teorem 3.1 to old, te constant in te error estimate depends exponentially on. Hence, in practice, a small enoug value sould be considered for ; see Section 5 for numerical experiments Piecewise affine finite elements: usual CFL condition. Te next teorem provides an a priori error estimate for p = 1 under te usual CFL condition. Teorem 3.2. Assume piecewise affine finite elements are used and tat u C 3 (0,T;L) C 0 (0,T;[H 2 (Ω)] m ). Ten, under te usual CFL condition (2.33) wit { min (2 2C L ) 2,(2 2C L C i) 2/3}, (3.23) al , version 1-4 May 2009 wit C i = C ic π and were C i is te constant in te inverse inequality v L d C i 1 v π 0 v L valid for all v V, tere olds u N u N L + ( N 1 n=0 1 8 τ un 2 S τ wn 2 S) 1/2 τ 2 + 3/2. (3.24) Proof. We bound ξ n+1 ζ n 2 L differently from te proof of Teorem 3.1. Set x n := ξn ζn, so tat (3.16) yields ξ n+1 ζ n = 1 2 τl x n τ(βn α n ). Hence, using a triangle inequality and Assumption (A5) yields ξ n+1 ζ n L 1 2 C Lστ x n L d C Lσ 1/2 1/2 τ x n S (3.25) τ βn α n L. Te first step is to control x n L d. Let yn = xn π0 xn and observe tat y n 2 L = (x n,y n ) L = τ(l ξ n,y n ) L τ(α n,y n ) L, since x n = τl ξ n ταn. To bound te first term in te rigt-and side, we use Assumption (A7) to infer τ (L ξ n,y n ) L C πσ 1/2 1/2 τ( ξ n S + ξ n L ) y n L. Furtermore, bounding te second term by a Caucy Scwarz inequality, using te CFL condition, and simplifying by y n L leads to Tus, y n L C πσ 1/2 1/2 τ ξ n S + τ α n L + Cτ 1/2 ξ n L. x n L d C i 1 y n L C iσ 1/2 3/2 τ ξ n S + C 1 (τ α n L + τ 1/2 ξ n L ),

15 RK-FEM WITH SYMMETRIC STABILIZATION 15 wit C i = C ic π. Hence, substituting back into (3.25) yields ξ n+1 ζ n L 1 2 C LC iσ 3/2 3/2 τ 2 ξ n S C Lσ 1/2 1/2 τ ξ n ζ n S + Cτ 1/2 (τ 5/2 + ξ n π + ζ n π + ξ n L ), were te contributions of α n and βn ave been bounded using (3.19) and te (generic) CFL condition (2.33). Owing to te condition (3.23), it is now inferred tat ξ n+1 ζ n L 2 5/2 τ 1/2 ( ξ n S + ζ n ξ n S ) + Cτ 1/2 (τ 5/2 + ξ n π + ζ n π + ξ n L ) Squaring yields ξ n+1 2 3/2 τ 1/2 ( ξ n S + ζ n S ) + Cτ 1/2 (τ 5/2 + ξ n π + ζ n π + ξ n L ). ζ n 2 L 3 8 τ ξn 2 S τ ζn 2 S + Cτ(τ 5 + ξ n π 2 + ζ n π 2 + ξ n 2 L). al , version 1-4 May 2009 Te conclusion readily follows by proceeding as in te proof of Teorem Analysis of explicit RK3 scemes. Tis section is devoted to te convergence analysis of explicit tree-stage RK3 scemes. We proceed similarly to Section 3. Te main difference is tat a sarper stability estimate can be derived for RK3 scemes, so tat te strengtened 4/3-CFL condition (2.34) is no longer needed. Finally, recall tat we assume ere u C 4 (0,T;L) and f C 3 (0,T;L) Tree-stage RK3 scemes. We consider scemes of te form wit te assumption tat w n = u n τl u n + τf n, (4.1) y n = 1 2 (un + w n ) 1 2 τl w n τ(fn + τ t f n ), (4.2) u n+1 = 1 3 (un + w n + y n ) 1 3 τl y n τψn, (4.3) ψ n = f n + τ t f n τ2 tt f n + δ n, δ n L τ 3. (4.4) Tere are many ways of writing explicit tree-stage RK3 scemes. Since te space differential operator is linear, tey all amount in te omogeneous case (f = 0) to u n+1 = u n τl u n τ2 L 2 u n 1 6 τ3 L 3 u n. (4.5) One example tat fits te above form is te tird-order Heun metod wic is usually written as follows: Straigtforward algebra yields ψ n = 5 4 fn fn+2/3 k 1 = L u n + f n, (4.6) k 2 = L (u n τk 1) + f n+1/3, (4.7) k 3 = L (u n τk 2) + f n+2/3, (4.8) u n+1 = u n τ(k 1 + 3k 3 ). (4.9) 1 2 τ tf n 3 2 τl (f n+1/3 f n 1 3 τ tf n ). (4.10) Proposition 4.1. Assume tat f C 2 (0,T;[H 1 (Ω)] m ) and tat s = s cip defined by (2.36) or tat s = s dg as defined by (2.38). Ten, (4.4) olds. as

16 16 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ Proof. We need to prove tat ψ n (fn + τ tf n τ2 tt f n) L τ 3. Set ψ n = A + B wit A = 5 4 fn fn+2/3 1 2 τ tf n and B = 3 2 τl (f n+1/3 f n 1 3 τ tf n). Using Taylor expansions yields A (f n + τ t f n τ2 tt f n ) L τ 3 f C3 (0,T;L). Consider now te term B and set z n := f n+1/3 f n 1 3 τ tf n so tat B = 3 2 τl (π z n ). Assumption (A5) yields L (π z n ) L z n L d + 1/2 π z n S, were we ave used te H 1 -stability of π in writing z n L d. Wen s = s dg, observe tat z n S = 0 since f C 2 (0,T;[H 1 (Ω)] m ) so tat 1/2 π z n S = 1/2 z n π z n S z n L d. Wen s = s cip, te boundary contribution is bounded as above, wile te interior contribution is bounded by a trace inequality and te H 1 -stability of π, yielding again 1/2 π z n S z n L d. As a result, using Taylor expansions yields al , version 1-4 May 2009 B L τ (f n+1/3 f n 1 3 τ tf n ) L d τ 3 f C2 (0,T;[H 1 (Ω)] m ), completing te proof Te error equation. Along wit definitions (3.10) and (3.11), let wit y = u + τ t u τ2 tt u. Lemma 4.1. Tere olds wit θ n = y n π y n, θ n π = y n π y n, (4.11) ζ n = ξ n τl ξ n + τα n, (4.12) θ n = 1 2 (ξn + ζ n ) 1 2 τl ζ n τβn, (4.13) ξ n+1 = 1 3 (ξn + ζ n + θ n ) 1 3 τl θ n τγn, (4.14) α n = L ξ n π, β n = L ζ n π, γ n = L θ n π π η n + δ n, (4.15) were η n = τ 1 I n 1 2 (t n+1 t) 3 tttt udt. Proof. Equations (4.12) and (4.13) are obtained as in Lemma 3.1. To derive (4.14), observe tat u n+1 = u n + τ t u n τ2 tt u n τ3 ttt u n τηn, and proceed again as in Lemma 3.1. For furter use, it is convenient to recall tat (4.12) (4.13) imply and to observe tat (4.13) (4.14) imply θ n = ζ n 1 2 τl (ζ n ξ n ) τ(βn α n ), (4.16) ξ n+1 = θ n 1 3 τl (θ n ζ n ) τ(γn β n ). (4.17)

17 RK-FEM WITH SYMMETRIC STABILIZATION Energy identity and stability. Our goal is now to derive an energy identity, ten leading to our main stability estimate. Lemma 4.2 (Energy identity). Tere olds 1 2 ξn+1 2 L 1 2 ξn 2 L τ ξn 2 S τ ζn 2 S τ θn 2 S θn ζ n 2 L = 1 6 τ ζn ξ n 2 S ξn+1 θ n 2 L + Λ n τ(γn,θ) n L τ(βn,ξ) n L τ(αn,ξ n ζn ) L, (4.18) were Λ n := 1 6 τ(λξn,ξn ) L 1 6 τ(λζn,ξn ) L τ(λθn,θn ) L. Remark 4.1. Te quantities 1 6 τ ζn ξn 2 S and 1 2 ξn+1 θ n 2 L appearing in te rigt-and side of te energy identity (4.18) are te anti-dissipative terms associated wit te explicit nature of te RK3 sceme. However, contrary to te RK2 sceme, tere is now a positive term in te left-and side of (4.18), namely 1 6 θn ζn 2 L, wic significantly improves te stability properties of te RK3 sceme, in particular circumventing te need for te strengtened 4/3-CFL condition for ig-order polynomials. Proof. Set A = 1 2 ξn+1 2 L 1 2 ξn+1 θ n 2 L 1 2 ξn 2 L. Ten, al , version 1-4 May 2009 A = (ξ n θn,θ) n L 1 2 ξn 2 L = 1 2 θn 2 L + (ξ n+1 θ,θ n ) n L 1 2 ξn 2 L = 1 2 θn 2 L 1 2 ξn 2 L 1 3 τ(l (θ n ζ),θ n ) n L τ(γn β,θ n ) n L, were (4.17) as been used. Set D 1 := 1 3 τ(γn β n,θ n ) L. Te term 1 2 θn 2 L 1 2 ξn 2 L can be evaluated using te energy identity (3.17) for te RK2 sceme upon replacing ξ n+1 by θ n. Setting yields D 2 := D τ(αn,ξ n ) L τ(βn,ζ n ) L τ(λξn,ξ n ) L τ(λζn,ζ n ) L, A = 1 2 τ ξn 2 S 1 2 τ ζn 2 S θn ζ n 2 L 1 3 τ(l (θ n ζ n ),θ n ) L + D 2 = 1 2 τ ξn 2 S 1 2 τ ζn 2 S 1 3 τ θn 2 S θn ζ n 2 L τ(l ζ n,θ n ) L + D 2 = 1 2 τ ξn 2 S 1 6 τ ζn 2 S 1 3 τ θn 2 S θn ζ n 2 L τ(l ζ n,θ n ζ n ) L + D 2 = 1 2 τ ξn 2 S 1 6 τ ζn 2 S 1 3 τ θn 2 S (θn ζ n,θ n ζ n τl ζ n ) L + D 2, wit D 2 := D τ(λθn,θn ) L and D 2 := D τ(λζn,ζn ) L. Consider te fourt term in te rigt-and side, say B, and observe tat owing to (4.12) and (4.16), B = 1 2 (θn ζ n, 1 3 (θn ζ n ) τl ξ n τ(βn α n )) L = 1 6 θn ζ n 2 L (θn ζ n,ξ n ζ n ) L τ(βn,θ n ζ n ) L. Set G = 1 2 τ ξn 2 S 1 6 τ ζn 2 S 1 3 τ θn 2 S 1 6 θn ζn 2 L and D 3 = D τ(βn,θn ζn ) L, so tat A = G (θn ζ n,ξ n ζ n ) L + D 3.

18 18 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ Ten, using again (4.16) leads to A = G ( 1 2 τl (ζ n ξ n ) τ(βn α n ),ξ n ζ n ) L + D 3 = G τ ζn ξ n 2 S + D 4, wit D 4 = D τ(βn αn,ξn ζn ) L 1 12 τ(λ(ζn ξn ),ζn ξn ) L. Using te expressions for D 1, D 2, and D 3 in D 4 yields (4.18). Lemma 4.3 (Stability). Under te usual CFL condition (2.33) wit tere olds min( C 1 S,(3 4 )1/2 C 1 L ), (4.19) ξ n+1 2 L ξ n 2 L τ ξn 2 S τ ζn 2 S τ θn 2 S Cτ(τ 6 + ξ n π 2 + ζ n π 2 + θ n π 2 + ξ n 2 L). (4.20) al , version 1-4 May 2009 Proof. We bound te terms in te rigt-and side of te energy identity (4.18). (i) Bound on 1 6 τ ζn ξn 2 S. Let ǫ and ˆǫ be positive real numbers to be cosen later. Observe tat ζ n ξ n 2 S (1 + ǫ) θ n ξ n 2 S + (1 + ǫ 1 ) θ n ζ n 2 S (1 + ǫ)(1 + ˆǫ) θ n 2 S + (1 + ǫ)(1 + ˆǫ 1 ) ξ n 2 S + (1 + ǫ 1 ) θ n ζ n 2 S (1 + ǫ)(1 + ˆǫ) θ n 2 S + (1 + ǫ)(1 + ˆǫ 1 ) ξ n 2 S + (1 + ǫ 1 )C S σ 1 θ n ζ n 2 L, were Assumption (A4) as been used. Ten, taking ǫ = 5 72 tat 1 6 (1 + ǫ)(1 + ˆǫ) = 7 24, 1 6 (1 + ǫ)(1 + ˆǫ 1 ) = 11 24, and tat 1 6 τ(1 + ǫ 1 )C S σ , owing to te coice (4.19) for te CFL condition, it is inferred tat (ii) Bound on 1 2 ξn τ ζn ξ n 2 S 7 24 τ θn 2 S τ ξn 2 S θn ζ n 2 L. θ n 2 L. Using (4.17) and te bound (2.31) yields 7 and ˆǫ = 11 and observing 1 2 ξn+1 θ n 2 L 1 9 τ2 L (θ n ζ) n 2 L τ2 γ n β n 2 L 1 9 (τc L σ 1 ) 2 θ n ζ n 2 L τ2 γ n β n 2 L 1 12 θn ζ n 2 L τ2 γ n β n 2 L, owing to te coice (4.19) for te CFL condition. (iii) Inserting te bounds delivered in steps (i) and (ii) into (4.18) yields 1 2 ξn+1 2 L 1 2 ξn 2 L τ ξn 2 S τ ζn 2 S τ θn 2 S 1 9 τ2 γ n β n 2 L τ(γn,θ n ) L τ(βn,ξ n ) L τ(αn,ξ n ζn ) L + Λ n. (iv) It remains to bound te five terms in te rigt-and side, say T 1 T 5. To tis purpose, we first bound te quantities α n, βn, and γn by proceeding as for te RK2 sceme. It is readily inferred tat τ α n L τ 1/2 ξ n π, τ β n L τ 1/2 ζ n π, τ γ n L τ 1/2 θ n π + τ 4,

19 RK-FEM WITH SYMMETRIC STABILIZATION 19 and tat for all v V, τ(α n,v ) L τ ξ n π ( v S + v L ), τ(β n,v ) L τ ζ n π ( v S + v L ), τ(γ n,v ) L τ θ n ( v S + v L ) + τ 4 v L. Moreover, still proceeding as for te RK2 sceme, ζ n L ξ n L + ξ π, θ n L ξ n L + ξ π + ζ n π. Using tese estimates yields T 1 τ(τ 7 + ζ n π 2 + θ n π 2 ). Furtermore, using Young inequalities leads to T 2 + T 3 + T τ ξn 2 S τ ζn 2 S τ θn 2 S + Cτ(τ 6 + ξ n π 2 + ζ n π 2 + θ n π 2 + ξ n 2 L). al , version 1-4 May 2009 Finally, since Λ is symmetric, T 5 τ( ξ n 2 L + ζ n 2 L + θ n 2 L) τ( ξ n π 2 + ζ n π 2 + ξ n 2 L). Collecting te above bounds and since τ 7 τ 6 concludes te proof Error estimate. Te next teorem provides a general a priori error estimate under te usual CFL condition. Teorem 4.1. Assume tat u C 4 (0,T;L) C 0 (0,T;[H p+1 (Ω)] m ). Under te CFL condition (2.33) wit te coice (4.19) for, tere olds u N u N L + ( N 1 n= τ un 2 S τ wn 2 S τ yn 2 S) 1/2 τ 3 + p+1/2. (4.21) Proof. Starting wit estimate (4.20), sum over n, apply Gronwall s lemma, and use te approximation property (2.32). 5. Numerical results. In tis section we investigate numerically te explicit RK2 and RK3 scemes using, respectively, teir implementations (3.6)-(3.8) and (4.6)- (4.9). Regarding space discretization, we consider te continuous (CIP stabilized) and discontinuous (DG) finite element metods discussed in 2.4, using piecewise affine (p = 1) and quadratic (p = 2) polynomials. We first illustrate te convergence properties of te various scemes by considering two test cases wit analytical smoot solutions. Ten, we investigate on a bencmark wit roug solution te capability of te present scemes to control spurious oscillations. Te numerical computations ave been carried out using FreeFem++ [21] Convergence rates for smoot solutions. We focus on two of te examples discussed in 2.1, namely te advection and acoustics equations. To illustrate te convergence rate of te discrete solutions, we ave computed te energy errors u N u N L at te final simulation time for specific sequences of τ- and refinements. Te scaling τ/ is cosen to satisfy te appropriate CFL condition and, according to te error estimates (3.22), (3.24) or (4.21), in suc a way tat te error in time dominates te error in space. More precisely, we ave taken:

20 20 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ For RK2 wit p = 1 and RK3 wit p = 2, τ = C (constant usual CFL); For RK2 wit p = 2, τ = C 4/3 (constant strengtened 4/3-CFL); For RK3 wit p = 1, τ = C 1/2. Tis coice is made in order to reduce te spatial error wic scales as 3/2 only. Te usual CFL increases as te time step is reduced; for te smallest time step considered, we ave ensured tat te scaling τ/ satisfies te required condition (4.19) Advection equation. As a first numerical test, we consider a twodimensional rotating Gaussian bencmark: we solve (2.10) wit β = (y,x) t, f = 0, Ω = {(x,y) R 2 : x 2 + y 2 1}, and te following Gaussian function, centered at te point (0.3,0.3), u 0 (x,y) = e 10[(x 0.3)2 +(y 0.3) 2], al , version 1-4 May 2009 as initial condition. Te stabilization parameter γ in S int F is set to 0.5 for DG (i.e., upwinding), and to and for CIP wit p = 1 and p = 2, respectively (improvements by furter tuning of tese parameters goes beyond te present scope; see, e.g., [4] for suc an investigation in te steady case). For eac numerical sceme, we report te energy errors at te final time T = 2π, i.e., after a complete rotation of te initial condition. Figure 5.1. Advection equation. Convergence istory for explicit RK2 wit continuous (CIP) and discontinuous (DG) affine finite elements (p = 1). (,τ) (,τ)/2 (,τ)/4 (,τ)/8 CIP DG Table 5.1 Advection equation. Elapsed CPU time (dimensionless) for te computation of te results reported in Figure 5.1.

21 RK-FEM WITH SYMMETRIC STABILIZATION 21 Figure 5.1 presents te convergence results for te RK2 sceme wit p = 1. We ave set τ/ = 0.2, wic amounts to coosing = 0.2 in te usual CFL condition (2.33) (recall tat, ere, σ = 1). Te RK2-CIP and te RK2-DG scemes exibit second-order accuracy in time, as stated in Teorem 3.2. On a fixed mes, te DG formulation yields more accurate results, togeter wit increased computational cost. Tis is reflected in Table 5.1 wic reports te corresponding (dimensionless) CPU times. Te increased accuracy and cost of te DG formulation can be related to te larger number of degrees of freedom. In tis sense, it is wort noticing tat all te linear systems ave been solved using te sparse direct solver provided wit FreeFem++ witout explicitly exploiting te block diagonal structure of te DG mass matrix (tis is particularly relevant in large-scale computations). Te convergence results for te RK2 sceme wit quadratic finite elements (p = 2) are presented in Figure 5.2. Here, te discretization parameters τ and satisfy te strengtened 4/3-CFL condition (2.34) wit = As expected, te RK2-CIP and RK2-DG scemes exibit te O(τ 2 ) accuracy predicted by Teorem 3.1. We also observe tat te strengtened 4/3-CFL condition seems to be numerically sarp. al , version 1-4 May 2009 Figure 5.2. Advection equation. Convergence istory for explicit RK2 wit continuous (CIP) and discontinuous (DG) quadratic finite elements (p = 2). Figure 5.3 presents te results for te RK3 sceme wit affine finite elements (p = 1). Here, τ scales as 1/2 and te usual CFL condition is satisfied wit an increasing parameter, up to Surprisingly, bot te RK3-CIP and te RK3- DG scemes sow a convergence rate iger tan te teoretical O(τ 3 ) stated in Teorem 4.1. A possible explanation is tat contributions of te spatial error can be dominant on te coarser meses. Finally, Figure 5.4 reports te results for RK3 wit quadratic finite elements (p = 2). Here, te usual CFL condition (2.33) olds wit = Bot te RK3-CIP and te RK3-DG scemes exibit te O(τ 3 ) accuracy stated in Teorem 4.1, altoug some perturbations, due to te O( 5/2 ) spatial contribution of error, are clearly visible on te finer meses Wave equation. We now consider te first-order PDE system (2.15) in te unit square Ω = [0,1] 2 wit reference velocity c 0 = 1 and final simulation time

22 22 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ al , version 1-4 May 2009 Figure 5.3. Advection equation. Convergence istory for explicit RK3 wit continuous (CIP) and discontinuous (DG) affine finite elements (p = 1). Figure 5.4. Advection equation. Convergence istory for explicit RK3 wit continuous (CIP) and discontinuous (DG) quadratic finite elements (p = 2). T = 1. Te rigt-and sides f 1 and f 2 and te initial data are cosen to yield te following exact solution: [ ] 1 p(x,y,t) = exp(t)sin(πx)sin(πy), q(x,y,t) = p(x,y,t). 1 For te DG metod, te free parameters in SF ext and SF int are set to γ 1 = 1 and γ 2 = γ 3 = 0.1, wile for te CIP metod, tey are set to γ 1 = 1 and γ 2 = γ 3 = 0.01 wit p = 1 and γ 2 = γ 3 = wit p = 2. In Figures 5.5 to 5.8, we report te

23 RK-FEM WITH SYMMETRIC STABILIZATION 23 convergence results for te energy norm of te error at final time. Te RK2 sceme wit affine finite elements (p = 1) exibits optimal time convergence, as sown in Figure 5.5. Here, te discretization parameters satisfy te usual CFL constraint wit = Te same optimal rate is obtained in Figure 5.6 wit quadratic finite elements (p = 2) and a set of time and space meses satisfying te strengtened 4/3-CFL condition wit = al , version 1-4 May 2009 Figure 5.5. Wave equation. Convergence istory for explicit RK2 wit continuous (CIP) and discontinuous (DG) affine finite elements. Figure 5.6. Wave equation. Convergence istory for explicit RK2 wit continuous (CIP) and discontinuous (DG) quadratic finite elements. As in te rotating Gaussian bencmark, a super-convergence beavior is obtained

24 24 E. BURMAN, A. ERN AND M. A. FERNÁNDEZ al , version 1-4 May 2009 Figure 5.7. Wave equation. Convergence istory for explicit RK3 wit continuous (CIP) and discontinuous (DG) affine finite elements. Figure 5.8. Wave equation. Convergence istory for explicit RK3 wit continuous (CIP) and discontinuous (DG) quadratic finite elements. for te RK3 sceme and affine finite elements (p = 1), see Figure 5.7. Te time step τ scales as 1/2 and te usual CFL is satisfied wit an increasing parameter, up to Finally, Figure 5.8 presents te results for RK3 wit quadratic finite elements (p = 2) and te usual CFL condition olding wit = Optimal O(τ 3 ) accuracy is obtained for te CIP and te DG discretizations.

25 RK-FEM WITH SYMMETRIC STABILIZATION 25 (a) Discrete initial condition (b) RK2-P 1 /Unstabilized Figure 5.9. Contour-lines for te discrete initial condition and te final discrete solution provided by te explicit RK2 sceme wit unstabilized continuous affine finite elements. al , version 1-4 May Controlling oscillations in roug solutions. For te advection bencmark discussed in 5.1.1, we now consider te following initial data: [ ( u 0 (x,y) = 1 e 10[(x 0.3)2 +(y 0.3) 2] ) ] 0.5 tan Tis function is smoot but as a sarp layer (wit tickness of order 0.001) leading to spurious oscillations wen using unstabilized continuous finite elements on meses tat are too coarse to resolve te internal layer. Our goal wit tis test case is to illustrate te capabilities of te metods analyzed in tis paper to control suc spurious oscillations. To tis aim, we consider a fixed uniform mes wit 256 elements along te boundary of Ω ( 0.025). Te sarp layer is tus under-resolved. Te discrete initial data takes te form of a cylinder of eigt 1 centered at te point (0.3,0.3); te contour-lines of its linear interpolant are sown in Figure 5.9a. Figure 5.9b sows te solution at final time, T = 2π (i.e., after one rotation of te initial data), obtained by te explicit RK2 sceme (1000 time steps) wit unstabilized continuous, piecewise affine finite elements (γ = 0 in te CIP metod). Te numerical solution is globally polluted by spurious oscillations. In Figure 5.10 we report te approximate solutions obtained wit te explicit RK2 sceme, using piecewise affine or quadratic CIP or DG finite elements and using te largest allowed (in terms of stability) time step size τ. As expected, global oscillations are eliminated by bot te CIP and te DG metods, and te numerical solution maintains its general aspect wit a sarp layer. Te increased accuracy of CIP and DG wit p = 2 is clearly visible. Note tat, compared to DG and for te considered values of γ, te CIP metod allows te use of larger time steps. Moreover, te critical value of τ depends on te polynomial order. In Figure 5.11 we ave finally reported te approximate solutions obtained wit te explicit RK3 sceme. Note te sligtly improved accuracy wit respect to Figure Once more, te critical value of τ depends on te polynomial order and te largest values were obtained wit te CIP metod (for te considered coice of te stabilization parameter). 6. Concluding remarks. In tis work, we ave analyzed several approximation metods to te evolution problem (1.1) combining explicit RK scemes in time and