REVISITING AND EXTENDING INTERFACE PENALTIES FOR MULTI-DOMAIN SUMMATION-BY-PARTS OPERATORS

Transcription

1 REVISITING AND EXTENDING INTERFACE PENALTIES FOR MULTI-DOMAIN SUMMATION-BY-PARTS OPERATORS Mark H. Carpenter, Jan Nordström David Gottieb Abstract A genera interface procedure is presented for muti-domain coocation methods satisfying the summationby-parts (SBP) spatia discretization convention. Unike more traditiona operators (e.g. FEM) appied to the advection-diffusion equation, the new procedure penaizes the soution and the first p derivatives across the interface. The combined interior/interface operators are proven to be pointwise stabe, and conservative, athough accuracy deteriorates for p. Penaties between two different sets of variabes are compared (motivated by FEM prima and fux formuations), and are shown to be equivaent for certain choices of penaty parameters. Extensive vaidation studies are presented using two casses of high-order SBP operators: 1) centra finite difference, and ) Legendre spectra coocation. summation-by-parts (SBP), high-order finite difference, finite eement, DG, numerica stabiity, interface conditions, conservation Appied and Numerica Mathematics Computationa Aerosciences Branch, NASA Langey Research Center, Hampton, VA Work performed in part, whie in residence at TU Deft, Deft, The Netherands. E-mai: mark.h.carpenter@nasa.gov. Computationa Physics Department, FOI SE-16490, Stockhom, Sweden; and the Department of Information Technoogy, Uppsaa, SE , Uppsaa, Sweden; The Department of Aeronautica and Vehice Engineering, KTH - The Roya Institute of Technoogy, SE-10044, Stockhom, Sweden; E-mai: Jan.Nordstrom@foi.se. Formery, Division of Appied Mathematics, Brown University, Providence, RI 091. David Gottieb passed away during the fina revision of this paper. He eft an indeibe mark on our ives, and wi be greaty missed. 1

2 1 Introduction High order finite difference methods (HOFDM) are ideay suited for simuations of compex physics requiring high fideity soutions, and for a simpe reason: efficiency. The extension of HOFDM to compex domains has proven difficut to accompish, however. Muti-domain HOFDM are natura candidates for extension, but are fraught with the compexity of interface couping procedures. Specificay, they must address the interface accuracy constraints, as we as contend with couped stabiity conditions and interface conservation issues. Many ad hoc procedures have been deveoped to accommodate muti-domain methods, incuding conforming, nonconforming, and overset/overapping procedures, (e.g. see Chesshire and Henshaw [10]). To enhance robustness, however, such techniques frequenty require reduction in the order of accuracy at the interface and/or incusion of speciaized dissipation terms. A simpe soution to the muti-domain probem for HOFDM appears in the work of Carpenter, Nordström, Gottieb[8] (henceforth referred to as CNG). Motivated by the recent popuarity/success of Discontinuous Gaerkin Finite Eement methods (DGFEM), and the interna penaty approaches, CNG combine summation-by-parts (SBP) operators [31, 6, 1, 7, 15, 16, 1, 3, 4, 33, 13] within each domain, with a penaty technique at the interface. The formuation requires ony weak grid continuity at the interface (C 0 : matching but not necessariy smooth), and is appicabe to any operator that satisfies a SBP property. It maintains forma accuracy through the interface, is conservative and maintains a bounded semi-discrete energy estimate consistent with the underying SBP operators. Simiar deveopments within the noda spectra eement community are reported by Hesthaven et a. [17, 18, 19]. Athough the CNG formuation is a powerfu technique for couping subdomains, it is not a genera approach. Indeed, penaties coud have been introduced in the origina work using any smooth, coocated interface variabe. A genera methodoogy is advantageous because it aows greater fexibiity in the construction of stabe, accurate and conservative interface operators. (For exampe, one target appication is the construction of bock interface couping operators needed for the newy deveoped energy stabe WENO (ESWENO) schemes [35].) Thus, our objective is to derive genera penaties between adjoining subdomains each discretized by a SBP operator. Herein, we present the framework for imposing penaties on the soution and first p interface derivatives. The new penaties are constructed by first parameterizing a possibe combinations of difference terms avaiabe at the interface. Energy methods are then used to constrain the parameters such that L and pointwise stabiity, and conservation are achieved at the interface. The newy derived penaties are formuated in two sets of variabes, reminiscent of the prima and fux formuations commony used in DGFEM for eiptic and paraboic equations. 1 Both formuations are shown to be consistent, stabe and conservative for the inear, constant coefficient, scaar advection diffusion equation. Furthermore, they are shown to be reated; i.e. the fux formuation can be impemented using the prima variabes if the interface penaties coefficients are aowed to vary with grid density. Numerous free parameters remain in both formuations even after stabiity and conservation constraints are imposed, and judicious choices for these parameters yied formuations anaogous to popuar schemes found in the FEM iterature. The construction of penaties from higher order derivatives is not without drawbacks. Indeed, it is shown that the forma accuracy suffers if terms invoving second derivatives are incuded in the penaties. Nevertheess, these terms are retained because they aow the construction of richer penaties, and because empirica evidence suggests that the theoretica accuracy estimates may not aways be sharp. This paper is organized as foows. In section, the genera approach for couping adjoining SBP operators is presented, that incudes penaties constructed from the soution and the first p derivatives. Next, it is shown that the penaties can be formuated in two distinct sets of variabes, athough the resuting formuations are cosey reated. Section 3 presents a derivation of the theoretica accuracy of the new penaties, in the context of HOFDM. Section 4 identifies specific vaues for the penaty parameters that produce schemes anaogous to severa commony used schemes. In section 5, numerica exampes are presented for a variety of SBP operators. A discussion and concusions are presented in sections 6 and 7, respectivey. 1 The fux formuation is impemented on a transformed set of equations obtained by expressing the second-order eiptic (paraboic) scaar equation as a system of first-order equations. The resuting interface variabes differ from those used in the conventiona prima approach. See Arnod et a. [3] for a detaied description and comparison of the two approaches. Note that both weak-form FEM approaches differ from the strong-form coocation approach commony used by HOFDM. See Carpenter et a. [9] for a comparison of weak- and strong-form approaches.

3 The semi-discrete probem.1 Definition of Stabiity Consider the continuous inear initia boundary vaue probem w t = Ew + F(x, t), x Ω, t 0, w = f(x), x Ω, t = 0, L C w = g(t), x Γ, t 0, (1) where E is the differentia operator and L C is the continuous boundary operator. The initia function f, the forcing function F, and the boundary term g are the data for the probem. Next, consider the semi-discrete counterpart of (1) (w j ) t = Ew j + F j (t), x j Ω, t 0, w j = f j, x j Ω, t = 0,, L D w j = g(t), x j Γ, t 0, () where E is the difference operator approximating the differentia operator E, L D is the discrete boundary operator, and f j, F j and g are the discrete initia function, forcing function, and boundary data, respectivey. Definition 1 The probem () is stabe, for F j (t) = g = 0 and a sufficienty fine mesh, if the soution w j satisfies where P is a suitaby defined discrete norm, such as a weighted L d dt w P 0 (3) w P = w T Pw. Note that this restrictive definition of stabiity does not aow for tempora soution growth, as woud be necessary to bound arbitrary source terms F j or boundary data g(t). It is, however, we suited for estabishing the stabiity of a numerica interface: our primary goa.. Discrete SBP Operators Define the P weighted inner product (v, w) P between two vectors, and et U and DU be the discrete approximations of the scaar quantities u and u x. The approximation DU of the first derivative satisfies the SBP rue DU = P 1 QU, Pu x Qu = PT e1, T e1 = O( x i, x b ) (4) (U, DV) P = U n V n U 0 V 0 (DU,V) P (5) (U,V) P = U T PV, P = P T, Q + Q T = B, B = diag[ 1, 0,.., 0, 1], (6) 0 < p min x I P p max x I. The matrix P is symmetric positive definite, the matrix Q is skew-symmetric with the exception of boundary points. A second derivative operator can be obtained by appying any two first derivative operators, or by constructing the operator directy. Both techniques are suitabe for spectra coocation. In HOFDM, using two consecutive nondissipative first derivatives, resuts in a second derivative operator that is unnecessariy wide and inaccurate and can Not to be confused with the vector norm, the nomencature P here refers to a suitabe matrix norm such as Frobenius. 3

4 ead to odd-even mode decouping. The operator, however, often eads to stabiity if varying coefficients are considered. Forming the HOFDM second derivative directy requires an operator with the foowing properties, D U = P 1 R U, Pu xx R u = PT e, T e = O( x i, x b ), (7) R = ( S T M + B)S, (8) as suggested in reference [8] (see aso [5]). The matrix B is given in (6), whie the matrix M must be positive definite. The matrix S is neary diagona with a discrete representation of the first derivative on the first and ast rows, where T e3 = O( x r ) and {Su} 0 = {Du} 0 = u x (x 0, t) + T e3, {Su} n = {Du} n = u x (x n, t) + T e3, S = 1 x s 00 s 01 s 0 s s nn 3 s nn s nn 1 s nn The second derivative defined in (7) and (8) satisfies a modified SBP rue. We have (U, D V ) P = U n {DV} n U 0 {DV} 0 (SU) T M(SV). The derivatives on the boundaries of S coincide with those found at the boundaries of the discrete operator D = P 1 Q. [See equation (4)]. The notation T e1, T e = O( x i, x b ) and T e3 = O( x r ) denotes that the approximation of the differentia operator is accurate to order i in the interior of the domain, to order b at the boundary and that the approximation of the boundary conditions is accurate to order r. The reation between the different orders of accuracy, i.e., i, b, r is discussed in section (4) beow, where forma proofs of accuracy are given for HOFDM...3 Genera Prima-Form Interface Penaties Consider the constant coefficient, inear Burgers equation u t + a u x = ǫu xx + F(x, t), t 0, 1 x 1, u(x, 0) = f(x), t = 0, 1 x 1, γu( 1, t) ǫu x ( 1, t) = g 1 (t) = L 1 (u), t 0, x = 1, ζu(+1, t) + ǫu x (+1, t) = g +1 (t) = L +1 (u), t 0, x = +1, (9) 0 a + ζ ; 0 a + γ. (10) with constants a and ǫ (0 < ǫ). A detaied description of the weposedness of this equation is omitted, but can be found in Carpenter et. a [9]. We are interested in the stabiity, conservation and accuracy of new interface treatments. For carity of presentation (and without oss of generaity), we sha negect the source terms, and the treatment of the physica boundary conditions in the subsequent derivations. This means that we wi consider stabiity of equations (9) and (10). Strong stabiity can easiy be obtained by a proper treatment of the outer boundary conditions, but is not the focus of this paper. We are mainy interested in the interface treatment. Assume that two adjoining SBP spatia operators support the first p derivatives on their respective domains. A coocation approximation of the equation (9), motivated by the FEM proposed by Baumann and Oden [4], that incudes penaties on the soution and the first p derivatives is given by P U t + ap D U = ǫp D D U + Σ p m=0 Σp k=0 mk[(d ku) i (Dr kv ) i] [D m ] T e i U(x, 0) = 0. P r V t + ap r D r V = ǫp r D r D r V + Σ p m=0 Σp k=0 r mk[(dr ku) i (Dr kv ) i] [Dr m]t e i+ V(x, 0) = 0. 4 (11)

5 x (n) = x r (0) x x (n ) x (n 1) x = 0 x r x r (1) x r () x r (3) Figure 1: The mesh cose to the interface at x = 0. The variabes in the eft (subscript ) [x (0) = 1, x (n) = 0] and right (subscript r) [x r (0) = 0, x r (nr) = +1] domains are Ū and V, respectivey, [see Figure (1)]. Definitions of the n (nr) dimensiona operators D (D r ) are given by D = P 1 Q ; 1 x 0 ; D r = Pr 1 Q r ; 0 x 1 and e i and e i+ are given by e i = [0, 0,..., 0, 1] T and e i+ = [1, 0,..., 0, 0] T. Note that any two SBP operators D, and D r woud suffice in equation (11). In fact, grid discontinuities ( x x r ), nonuniform grids, or even operators of different order or type coud be used. 3 What remains is to determine the vaues of the interface parameters 00,, pp, r 00, r pp that ead to stabiity, conservation and accuracy..3.1 Stabiity To faciitate the derivation of precise stabiity bounds for the interface parameters, we confine our discussion to the cases p ; i.e. ony the soution and the first two derivatives are penaized at the interface. In this case, equation (11) simpifies to P U t + ap D U = ǫp D D U + { 00 [U i V i ] + 01 [(D 1 U) i (D 1 rv ) i ] + 0 [(D U) i (D rv ) i ]} [I ] T e i + { 10 [U i V i ] + 11 [(D 1 U) i (D 1 rv ) i ] + 1 [(D U) i (D rv ) i ]} [D ] T e i + { 0 [U i V i ] + 1 [(D 1U) i (DrV 1 ) i ] + [(D U) i (DrV ) i ]} [D ]T e i P r V t + ap r D r V = ǫp r D r D r V (1) + {r 00 [V i U i ] + r 01 [(D 1 r V ) i (D 1 U) i] + r 0 [(D r V ) i (D U) i]} [I r ] T e i+ + {r 10 [V i U i ] + r 11 [(D 1 rv ) i (D 1 U) i] + r 1 [(D rv ) i (D U) i]} [D r ] T e i+ + {r 0 [V i U i ] + r 1 [(D 1 r V ) i (D 1 U) i] + r [(D r V ) i (D U) i]} [D r ]T e i+ The vaues of the interface parameters 00, 01, 0, 10, 11, 1, 0, 1,, and r 00, r 01, r 0, r 10, r 11, r 1, r 0, r 1, r, that resut in a stabe interface are given in the foowing theorem. Theorem 1 The approximation (1) of the probem (9) is stabe if the eighteen parameters are reated by the six equaities 00 = r 00 + a ; 01 = r 01 ǫ r 0 = 0 ; r 0 = 0 ; r 1 = 1 ; r = (13) and constrained by the foowing inequaities [r ] [(α + β)ǫ] (14) [ǫ( α + β) (r )] ǫ(α + β) (r ) [(α + β)ǫ] (15) (+r 1+r ) 4[ǫ(α+β) (r {( [ǫ( α+β) (r 1 r 1 ) )](+r 1 +r ) } [ǫ(α+β) (r )] )] 4{[(α+β)ǫ] [ǫ( α+β) (r )] } [ǫ(α+β) (r )] (16) 3 Even materia interfaces between adjoining media are aowed (see Nordströ and Gustafsson [7]). 5

6 00 a [t 1 +t ] 8[(α+β)ǫ][s 1 +s [t [s 1 t + 1 s ][t 1 +t ] ] (s 1 +s ) ] 4(α+β)ǫ[1 [s 1 s ] [s 1 +s ] ] with and { x 1 + (t 1 +t )(y 1 +y ) (α+β)ǫ(s 1 +s ) + [t 1 t + 4{ + (s 1 s )(t 1 +t ) (s 1 +s ) [y 1 +y ] 8[(α+β)ǫ][s 1 +s ] + [y 1 y + ][y 1 y + (s 1 s )(y 1 +y ) ] (s 1 +s ) } 4(α+β)ǫ[1 [s 1 s ] [s 1 +s ] ] [s 1 s ][y 1 +y ] ] (s 1 +s ) } 4(α+β)ǫ[1 [s 1 s ] [s 1 +s ] ] t 1 = (ǫ ) ; t = ( 01 + r 10 ) r 11 = ǫ[( α + 3β)/4 s 1 (α + β)] ; 11 = ǫ[(3α β)/4 s (α + β)], x 1 = (r ) ; y 1 = ( 1 + r 1 ) ; y = (r 1 + r 1 ) (17) (18) 1 1 α = ; β = e T i P 1 e i e T i+ Pr 1 e i+ (19) Proof : Stabiity of (1) foows if the interface treatment at x = x i is of a dissipative nature. The energy method appied to equation (1) (mutipying the discrete equations in the eft and right subdomains by U T and V T respectivey, and adding the resut to its transpose), and using the reation Q + Q T = B from the SBP rue (6), yieds d dt [ U P + V P r ] + ǫ[ D U P + D r V P r ] = [ ǫud U a U ] i + [ ǫv D r V a V ] i+ + U i { 00 [U i V i ] + 01 [(D U) i (D r V ) i ] + 0 [(D U) i (Dr V ) i]} + (D 1U) i { 10 [U i V i ] + 11 [(D U) i (D r V ) i ] + 1 [(D U) i (DrV ) i ]} + (D U) i { 0 [U i V i ] + 1 [(D U) i (D r V ) i ] + [(D U) i (Dr V ) i]} + V i {r 00 [V i U i ] + r 01 [(D r V ) i (D U) i ] + r 0 [(DrV ) i (D U) i]} + (DrV 1 ) i {r 10 [V i U i ] + r 11 [(D r V ) i (D U) i ] + r 1 [(DrV ) i (D U) i]} + (Dr V ) i {r 0 [V i U i ] + r 1 [(D r V ) i (D U) i ] + r [(Dr V ) i (D U) i]} (0) Note again that we have negected the outer boundary terms. A sma portion of the diffusion terms D U P and D r V Pr arising at the interface can be moved to the (RHS) to hep in the proof of stabiity. Defining P = P α e i e i T ; Pr = P r β e i+ e i+ T (1) with α and β as defined in equation (19) and coecting a the interface terms, equation (0) becomes ] ] [ U P + V Pr + ǫ [ D U P + D r V Pr = Ῡi () d dt with the interface term Ῡi = [J i ] T J i Mi J i defined by = [ U i, V i, (D U) i, (D r V ) i, (D U) i, (D r V ) i ] T (3) M i = ( a + 00 ) ( 00 + r 00 ) (ǫ ) ( 01 + r 10 ) ( ) ( 0 + r 0 ) ( 00 + r 00 ) (+a + r 00 ) (r ) ( ǫ + r 01 + r 10 ) ( 0 + r 0 ) (r 0 + r 0 ) (ǫ ) (r ) ( 11 αǫ) ( 11 + r 11 ) ( ) ( 1 + r 1 ) ( 01 + r 10 ) ( ǫ + r 01 + r 10 ) ( 11 + r 11 ) (r 11 βǫ) ( 1 + r 1 ) (r 1 + r 1 ) ( ) ( 0 + r 0 ) ( ) ( 1 + r 1 ) ( + r ) ( 0 + r 0 ) (r 0 + r 0 ) ( 1 + r 1 ) (r 1 + r 1 ) ( + r ) r (4) The origins of equation (19) are derived in reference [9] incuding a proof of the definiteness of P and P r. 6

7 Stabiity of the interface is guaranteed if the matrix M i is negative semi-definite; i.e. Ῡ i = [J i ] T Mi J i 0. The definiteness of a symmetric matrix is governed by the sign of its eigenvaues. Thus, Syvester s Theorem is used to systematicay rotate M i into diagona form whie maintaining the signs of its eigenvaues. The first rotation of Mi is the foowing simiarity transformation. Define the new vector Ĵi = SJ i such that: U i + V i U i Jˆ i = 1 U i V i (D U) i + (D r V ) i (D U) i (D r V ) i = V i (D U) i (D U) i + V ) (D r V ) i (5) i (D (D U) i (DrV U) i ) i (DrV ) i The rotation matrix S repaces the stabiity condition given in equation (4) with the foowing equivaent condition: Ji T M i J i = Ji T S T S M i S T SJ i = ˆ i J T ˆM i ˆ Ji 0. (6) The rotated matrix ˆM i has zeros on the main diagona. (See reference [9] for detais.) Thus, definiteness of ˆMi requires a off-diagona terms in the respective rows be zero, and produces the generaized conservation conditions 00 = r 00 + a ; 01 = r 01 ǫ r 0 = 0 ; r 0 = 0 ; r 1 = 1 ; r = The necessary agebra is greaty simpified through a change of variabes. Defining t 1 = (ǫ ) ; t = ( 01 + r 10 ) r 11 = ǫ[( α + 3β)/4 s 1 (α + β)] ; 11 = ǫ[(3α β)/4 s (α + β)], x 1 = (r ) ; y 1 = ( 1 + r 1 ) ; y = (r 1 + r 1 ) (7) yieds the reduced matrix M i = a t 1 t t 1 + t 0 x 1 0 t 1 t ((α + β)ǫ) (α + β)ǫ(s1 s) 0 y 1 y 0 t 1 + t (α + β)ǫ(s1 s) (α + β)ǫ(s1 + s) 0 y 1 + y x 1 y 1 y y 1 + y 0 4 (8) Repeated appication of Syvester s Theorem produces the stabiity inequaities 0 [s 1 + s ] 0 1 [s 1 s ] [s 1 +s ] [y 1 +y ] 8[(α+β)ǫ][s 1 +s [y [s 1 y + 1 s ][y 1 +y ] ] (s 1 +s ) ] 4(α+β)ǫ[1 [s 1 s ] [s 1 +s ] ] 00 a [t 1 +t ] 8[(α+β)ǫ][s 1 +s [t [s 1 t + 1 s ][t 1 +t ] ] (s 1 +s ) ] 4(α+β)ǫ[1 [s 1 s ] [s 1 +s ] ] (9) which are the desired resut. { x 1 + (t 1 +t )(y 1 +y ) (α+β)ǫ(s 1 +s ) + [t 1 t + 4{ + (s 1 s )(t 1 +t ) (s 1 +s ) [y 1 +y ] 8[(α+β)ǫ][s 1 +s ] + [y 1 y + 7 ][y 1 y + (s 1 s )(y 1 +y ) ] (s 1 +s ) } 4(α+β)ǫ[1 [s 1 s ] [s 1 +s ] ] [s 1 s ][y 1 +y ] ] (s 1 +s ) } 4(α+β)ǫ[1 [s 1 s ] [s 1 +s ] ]

8 Remark. As mentioned earier we can easiy extend this resut to strong stabiity by incuding the outer boundary conditions. Remark. A genera interface procedure incuding derivative terms up to p th -order and satisfying an L stabiity condition, woud require simiar inequaities for a interface parameters 00 pp, and r 00 r pp. Such inequaities coud in principe be derived using a simiar approach, but are not presented herein..3. Pointwise Stabiity Three distinct groups of terms appear in equation (). The groups incude the soution terms U P and V P r, the derivative terms D U P and D r V Pr, and the interface term Υ i. The soution and derivative terms are weaky bounded in P-norms for finite times, whie the interface term is ocay pointwise bounded. Note that equation () does not estabish the pointwise stabiity of the soution at interior gridpoints. To estabish interior pointwise boundedness, a discrete Soboev inequaity is required. A proof of the discrete Soboev inequaity, (U i ) c 1 U P + c DU P derived specificay for singe-domain, SBP operators, is presented in reference [9]. This inequaity, combined with equation (), provides the ink between P-boundedness of U and DU, and pointwise boundedness of the soution over the entire domain, and ead to the foowing theorem. Theorem For any grid function U on (0 x j );j = 0, 1,,n, and a consistent derivative operator D of rank n 1, then P-boundedness of U P and DU P, impies a pointwise estimate of the form U (c mpm + 1 ) U + c mpm 1 DU (30) Proof : Refer to reference [1] for the proof of Theorem (). Extension to two domains is straightforward. Remark. In addition to providing a stronger measure of stabiity, a pointwise stabiity estimate can be a necessary condition used in proofs of goba accuracy. Indeed, a pointwise estimate is the critica condition used in the work of Svärd and Nordström [34], and in Section 4 of this work, to estabish a sharp estimate of goba soution accuracy. Remark. A consistent derivative operator D of rank n 1, is needed to extend the cassica proofs[1], deveoped using first-order difference operators, to those using high-order differences..3.3 Prima Conservation We are interested in numerica soutions to the 1-D equation in divergence form u t + f x = 0, x 1, t 0 ; f = a u ǫ u x (31) where f invoves convective and diffusive terms arising from for exampe the inear Burgers equation. Consider the specia case for which f is discontinuous (i.e. f x is undefined). Then equation (31) must be repaced by the reated weak statement 1 1 Φ(x, t)u(x) T 0 dx + T where Φ(x, t) is an arbitrary smooth function. 0 Φ(x, t)f(t) +1 1 dt = T [Φ t (x, t)u(x, t) + Φ x (x, t)f(u(x, t))]dxdt (3) Semi-discrete operators must satisfy an additiona constraint: conservation, if they are to accuratey resove discontinuous data. Discrete conservation can be defined in many ways, but herein it is defined as foows: Definition Assume that the discrete soution converges to a function u(x, t). If this convergent imit function u(x, t) is a weak soution to equation (31) that satisfies equation (3), then the semi-discretization is said to be conservative. In practica terms, semi-discretize operators used for equation (31) must aow for discrete manipuation to a form that is discretey equivaent to equation (3): i.e. spatia integras repaced with discrete quadratures, and must converge 8

9 to equation (3) in the imit of infinite resoution. As wi be shown, this is a deicate process at interna boundaries between subdomains. A variant of the cassica Lax-Wendroff Theorem [] is now presented in the context of the SBP formuation defined in equation (1). Specia attention is given to the interface conditions, and the diffusive terms. For ease of presentation, we sha confine our attention to the case p 1, i.e. ony the soution and the first derivative are penaized across the interface. Theorem 3 Two conditions are necessary if approximation (1) is to be conservative (in the P norm) across the interface. They are 00 = r 00 + a, 01 = r 01 ǫ, (33) Proof : Mutipying (1) with Φ T P and Φ T r P r, (negecting the farfied boundary terms and assuming for carity that p 1) eads to Φ T P U t + Φ T r P r V t + a[φ T P D U + Φ T r P r D r V] ǫ[ Φ T P D D U + Φ T r P rd r D r V] = 00 Φ i [U i V i ] + 01 Φ i [(D U) i (D r V ) i ] + 10 (Φ ) i [U i V i ] + 11 (Φ ) i [(D U) i (D r V ) i ] + r 00 Φ i [V i U i ] + r 01 Φ i [(D r V ) i (D U) i ] + r 10 (Φ ) i [V i U i ] + r 11 (Φ ) i [(D r V ) i (D U) i ] (34) where Φ i and (Φ ) i are the vaues of the test function and its derivative at the interface ocation x i. Both are sufficienty continuous across the interface. Note the ambiguity in the meaning of conservation for the diffusive terms. For exampe, using the SBP rues [(5) (6)] once on the diffusion terms yieds, Φ T P D D U + Φ T r P rd r D r V = (Φ )T P D U) (Φ r )T P r D r V ) + [Φ (D U)] i 1 + [Φ r (D r U)] +1 i +. (35) Rearranging the time terms, and using the SBP rues [(5) (6)] on the convection terms in equation (34), then simpifying using equation (35) produces the conservation equation d dt (ΦT P U + Φ T r P rv) +[Φ (a ǫd )U] 1 + [Φ r (a ǫd r )V ] +1 = [(Φ ) T t P U + (Φ r ) T t P rv] + (Φ ) T P (ai ǫd )U + (Φ r) T P r (ai r ǫd r )V + Φ i ( 00 r 00 a)(u i V i ) + Φ i ( 01 r 01 + ǫ)(du i DV i ) + (Φ ) i ( 10 r 10 )(U i V i ) + (Φ ) i ( 11 r 11 )(DU i DV i ) (36) Integrating equation (36) with respect to time, appying the conservation conditions given by equation (33) and the additiona conditions 10 = r 10 and 11 = r 11 produces the equation (Φ T P U + Φ T r P r V) T 0 + T 0 {[Φ (a ǫd )U] 1 + [Φ r (a ǫd r )V ] +1 }dt = T 0 [(Φ ) T t P U + (Φ r ) T t P rv]dt + T 0 {(Φ ) T P (ai ǫd )U + (Φ r) T P r (ai r ǫd r )V}dt (37) which matches the weak soution given by equation (3) term by term, in the imit of infinite spatia resoution. Using the SBP rues [(5) (6)] twice on the diffusion terms yieds, Φ T P D D U + Φ T r P r D r D r V = (Φ )T P U) + (Φ r) T P r V ) + [Φ (D U) (Φ )U] i 1 + [Φ r (D r U) (Φ (38) r )V ] +1 i + 9

10 which eads to the resuting conservation equation d dt (ΦT P U + Φ T r P rv) +[Φ (a ǫd )U] 1 + [Φ r (a ǫd r )V ] +1 = [(Φ ) T t P U + (Φ r ) T t P rv] + (Φ ) T P (ai )U + (Φ r) T P r (ai r )V + ǫ(φ ) T P U + ǫ(φ r) T P r V ǫ[φ U 1 + Φ r V +1 ] + Φ i ( 00 r 00 a)(u i V i ) + Φ i ( 01 r 01 + ǫ)(du i DV i ) + (Φ ) i ( 10 r 10 ǫ)(u i V i ) + (Φ ) i ( 11 r 11 )(DU i DV i ) (39) Integrating equation (39) with respect to time, appying the conservation conditions given by equation (33) and the additiona conditions 10 = r 10 + ǫ and 11 = r 11, yieds an expression simiar to equation (37) (see reference [9]) that matches the weak form of the advection diffusion equation in the imit of infinite spatia resoution. Remark. Note that the diffusive constraint conditions for the variabe Φ in equations (36) and (39) depend on the choice of definition for conservation. Sti another definition (perhaps a inear combination of the two) woud yied a third set of constraints. This arbitrariness probaby means that diffusive conservation is not as important as convective conservation..4 A Fux-Form Interface Penaty An SBP interface penaty (restricted for carity to p 1) that approximates equation (9), motivated by the Loca Discontinuous Gaerkin (LDG) FEM proposed by Cockburn and Shu [11] is P U t + ap D U ǫp D φ = L 00 [U i V i ] e i + L 01 [(φ) i (ψ) i ] e i ǫp (φ D U) = L 10 [U i V i ] e i + L 01 [(φ) i (ψ) i ] e i U(x, 0) = 0. P r V t + ap r D r V ǫp r D r ψ = R 00 [V i U i ] e i+ + R 01 [(ψ) i (φ) i ] e i+ ǫp r (ψ D r V) = R 10 [V i U i ] e i+ + R 11 [(ψ) i (φ) i ] e i+ V(x, 0) = 0. (40) Theorem 4 The approximation (40) of the probem (9) is stabe if the eight parameters are reated by the two equaities L 00 = R 00 + a ; L 01 = R 01 ǫ (41) and constrained by the foowing inequaities [R 11 + L 11 ] [(α + β)ǫ] (4) [ǫ( α + β) (R 11 L 11 )] ǫ(α + β) (R 11 + L 11 ) [(α + β)ǫ] (43) L 00 a (ǫ+l 01+L 10 +R 10 ) 4[ǫ(α+β) (R 11 +L 11 {ǫ+l [ǫ( α+β) (R 10 R L 11 )](ǫ+l 01 +L 10 +R 10 ) } [ǫ(α+β) (R 11 +L 11 )] )] 4{(α+β)ǫ [ǫ( α+β) (R 11 L 11 )] } [ǫ(α+β) (R 11 +L 11 )] where (as with the prima form) (44) 1 1 α = e T ; β = i P 1 e i e T i+ Pr 1 e i+ Proof : The stabiity proof for the LDG scheme is omitted, as it is equivaent to that used for the prima scheme. 10

11 Remark. Athough the stabiity boundaries for the eight interface parameters are identica for the prima and LDG schemes, as are the norms used in both cases, the two methods are stabe in different combinations of terms: DU vs. φ. Remark: A genera prima interface penaty that incudes derivatives up to p th -order, woud require constraints on the interface parameters L 00 L pp, and R 00 R pp. A stabiity anaysis coud be used to identify conservation conditions, as we as the parameter space that guarantees L stabiity. Interior pointwise boundedness is estabished using a discrete Soboev inequaity. This inequaity, (see reference [9] or [1]) combined with theorem (4), provides the ink between P-boundedness of U and φ, and pointwise boundedness of the soution over the entire domain, and ead to the foowing theorem. Theorem 5 For any grid function U on (0 x j ) ; j = 0, 1,,n, a consistent derivative operator D of rank n 1, and φ as defined in equation (40), then P-boundedness of U P and φ P, impies a pointwise estimate of the form U (c mpm + 1 ) U + c 1 mpm φ (45) Proof : Refer to reference [1] for a proof of Theorem (5)..5 Reating the Prima and Fux Formuations One might imagine that the prima and fux formuations are equivaent (as shown in FEM by Arnod et a [, 3]) with an appropriate change of interface variabes. We begin by showing that the LDG formuation can be impemented in terms of the prima scheme if specific vaues of the interface parameters are used. For ease of presentation, we assume that ony the soution and first derivatives are penaized across the interface. Theorem 6 The LDG interface parameters L 00 - R 11 are reated to the prima interface parameters 00 - r 11 by the foowing reations: with 00 = L 00 + L 01 c 1 + α + L11c1 α 01 = 0 + L 01 c L11c α 10 = L 10 L 11 c 1 11 = L 11 c R r 00 = R 00 + R 01 c 1 10 β R11c1 β r 01 = 0 + R 01 c + 0 R11c β r 10 = R 10 R 11 c 1 r 11 = R 11 c c 1 = 1 ǫ ( L 10 α + R 10 β ) 1 1 ǫ ( L 11 α + R 11 L 10 1 ; c = β ) 1 1 ǫ ( L 11 α + R 11 (46) β ) (47) Proof : The LDG two-domain formuation is given in equation (40). Soving equation (40) for φ and ψ yieds the expressions φ = D U + 1 ǫ P 1 {L 10 [U i V i ] e i + L 11 [(φ) i (ψ) i ] e i } ψ = D r V + 1 (48) ǫ P 1 r {R 10 [V i U i ] e i+ + R 11 [(ψ) i (φ) i ] e i+ } whie, soving equation (48) for the interface vaues yieds the expressions φ i = e T i φ = (D U) i + 1 ǫα {L 10[U i V i ] + L 11 [(φ) i (ψ) i ]} ψ i = e T i+ ψ = (D r V) i + 1 ǫβ {R 10[V i U i ] + R 11 [(ψ) i (φ) i ]} (49) with the difference being φ i ψ i = (D U) i (D r V) i + 1 ǫ (L 10 α + R 10 β )[U i V i ] + 1 ǫ (L 11 α + R 11 β )[φ i ψ i ] (50) 11

12 Rewriting equation (50) in terms of the difference φ i ψ i yieds φ i ψ i = c 1 [U i V i ] + c [(D U) i (D r V) i ] c 1 = 1 ǫ ( L 10 α + R 10 β ) 1 1 ǫ ( L 11 α + R 11 β ) ; c = ǫ ( L 11 α + R 11 β ) (51) Substituting equations (48), (49) and (51) back into the origina LDG formuation [equation (40)] yieds the expression P U t + ap D U = ǫp D D U + {L 00 [U i V i ] + L 01 (c 1 [U i V i ] + c [(D U) i (D r V) i ])} I e i + {L 10 [U i V i ] + L 11 (c 1 [U i V i ] + c [(D U) i (D r V) i ])} P r V t + ap r D r V = ǫp r D r D r V P D P 1 e i + {R 00 [V i U i ] + R 01 (c 1 [V i U i ] + c [(D r V) i (D r U) i ])} I r e i+ + {R 10 [V i U i ] + R 11 (c 1 [V i U i ] + c [(D r V) i (D r U) i ])} P r D r Pr 1 e i+ Simpifying equation (5) using the reations (5) P D P 1 e i = D T e i + 1 α e i P r D r P 1 r e i+ = D T r e i 1 β e i + yieds P U t + ap D U ǫp D D U = [ + [ + [ + [ P r V t + ap r D r V ǫp r D r D r V = [ + [ + [ + [ L 00 + L 01 c 1 L 10 α + L11c1 α ] [U i V i ] I e i 0 + L 01 c ] [(D 0 + L11c U) i (D r V) i ] I e i α ] [U L 10 L 11 c i V i ] D Te i ] [(D 0 L 11 c U) i (D r V) i ] D Te i R 00 + R 01 c 1 ] [V R10 β R11c1 i U i ] I r e i+ β 0 + R 01 c 0 R11c β ] [(D r V) i (D r U) i ] I r e i+ ] [V R 10 R 11 c i U i ] Dr T e i ] [(D 0 R 11 c r V) i (D r U) i ] Dr T e i+ (53) Comparing equation (53) with the origina prima scheme given in equation (1) gives the desired resut. Remark. Note that the prima coefficients are not constants, but rather change with grid resoution. For exampe, the new parameter 00 = L 00 + L 01 c 1 + L 10 α + L 11c 1 α given in equation (46) is inversey proportiona to the grid spacing x since α x, and becomes arger as the grid is refined. It is reasonabe to ask whether the stabiity constraints (9) for the prima method are preserved for methods satisfying the LDG stabiity constraints (4) and (43). Theorem 7 Stabiity (in the P-norm) in the LDG variabes is a sufficient condition for stabiity (in the P-norm) in the prima variabes. Proof : We begin by assuming that the LDG variabes L 00 R 11 are chosen such that the conservation conditions L 00 = R 00 + a, L 01 = R 01 ǫ, 1

13 are satisfied, and that vaues of the parameters S 1 and S satisfy the constraint equations 0 [S 1 + S ] ; [S 1 S ] [S 1+S ] 1 and L 00 a [T 1 + T ] 8[(α + β)ǫ][s 1 + S ] [T 1 T + [S 1 S ][T 1+T ] (S 1 +S ] ) 4(α + β)ǫ[1 [S 1 S ] ] [S 1 +S ] Substitution of the transformation reations given by equation (46) into the prima conservations conditions 00 = r 00 + a ; and 01 = r 01 ǫ, yieds the conservation equations based on the LDG variabes; 00 r 00 a = 0 (L 00 R 00 a) + 4(αR 10+βL 10 )(L 01 R 01 +ǫ) ǫ[(α β) +4(α+β)(αS 1 +βs = 0 )] 4(αβ)(L 01 r 01 + ǫ = 0 01 R 01 +ǫ) [(α β) +4(α+β)(αS 1 +βs = 0 )] (54) whie substitution of the transformation reations given by equation (46) into the prima and the stabiity constraints given by (9) yieds the stabiity equations based on the LDG variabes; 0 [s 1 + s ] 0 [α (1+4S 1 ) β (1+4S )] +64α β [S 1 +S ][1 (S 1 S ) (S 1 +S ) ] [((α β) +4(α+β)(αS 1 +βs ))] 0 [1 (s 1 s ) ] 0 3α β (S [S 1 +S ][1 1 S ) (s 1 +s ) (S 1 +S ) ] [α (1+4S 1 ) β (1+4S )] (55) and 00 a L 00 a [t 1 +t ] 8[(α+β)ǫ][s 1 +s [t [s 1 t + 1 s ][t 1 +t ] ] (s 1 +s ) ] 4(α+β)ǫ[1 [s 1 s ] [s 1 +s ] ] [T 1 +T ] 8[(α+β)ǫ][S 1 +S [T [S 1 T + 1 S ][T 1 +T ] ] (S 1 +S ) ] 4(α+β)ǫ[1 [S 1 S ] [S 1 +S ] ] Thus, if the origina scheme written in the LDG variabes satisfies the required conservation and stabiity constraints, then it aso satisfies the corresponding conservation and stabiity constraints written in the prima variabes. Remark: It has not been estabished whether stabiity (in the P-norm) in the LDG variabes is a necessary condition for stabiity (in the P-norm) in the prima variabes. (56) 3 Forma Accuracy We derive error estimates for the semi-discrete discretization method given by equation (1), under the assumption that p 1. Extensive anaysis of the accuracy of FEM is avaiabe in the iterature (see for exampe the work of Arnod et a. [3]). Thus, the focus of attention herein is high-order finite difference methods. Define the vectors u = [u(x 0, t),, u(x n, t)] T, u x = [u x (x 0, t),, u x (x n, t)] T, and u xx = [u xx (x 0, t),,u xx (x n, t)] T, to be the projected vaues of the exact soution and derivatives, defined in the eft domain at gridpoints x. Simiary, define the vectors v, v x and v xx in the right domain. With these definitions, the continuous probem (9) can be represented in vector nomencature as u t + au x ǫu xx = 00 [u i v i ] P 1 I e i + 01 [u x v x ] i P 1 I e i + 10 [u i v i ] P 1 D Te i + 11 [u x v x ] i P 1 D Te i u(x, 0) = 0. v t + av x ǫv xx = r 00 [v i u i ] Pr 1 I r e i+ + r 01 [v x u x ] i Pr 1 I r e i+ + r 10 [v i u i ] P 1 r D T r e i+ + r 11 [v x u x ] i P 1 r D T r e i+ v(x, 0) = (57)

14 with zero initia data, negected outer boundary conditions and source term F(x, t). Note that the interface terms [u i v i ] and [u x v x ] i are identicay zero for the exact soution, but have been added to the right hand side of equation (57) to faciitate subsequent manipuations. Substituting the discrete accuracy reations u x = D u + T e1 and u xx = D D u + T e [see definitions (4) and (7)], into equation (57) yieds I e i D Te i I e i + ǫt e + 11 [(T e1 ) i (T e1r ) i ] P 1 D T e i u t + ad u ǫd D u = 00 [u i v i ] P 1 I e i + 01 [(D u) i (D r v) i ] P [u i v i ] P 1 D Te i + 11 [(D r v) i (D u) i ] P 1 at e [(T e1 ) i (T e1r ) i ] P 1 u(x, 0) = 0. v t + ad r v ǫd r D r v = r 00 [v i u i ] Pr 1 I r e i+ + r 01 [(D r v) i (D u) i ] Pr 1 I e i + r 10 [v i u i ] Pr 1 Dr Te i + + r 11 [(D u) i (D r v) i ] Pr 1 D Te i at e1r + r 01 [(T e1r ) i (T e1 ) i ] Pr 1 I r e i + ǫt er + r 11 [(T e1r ) i (T e1 ) i ] Pr 1 Dr T e i v(x, 0) = 0. (58) Next, define the semi-discrete error vectors Ξ = U u and Ξ r = V v in the eft and right domains, respectivey. Pre-mutipying by P 1 and subtracting equations (57) and (58) produces an evoution equation for the error vectors. Specificay, the coocation error of equation (9) written in prima form is Ξ t + ad Ξ ǫd D Ξ = 00 [Ξ i Ξ ri ] P 1 I e i + 01 [(D Ξ ) i (D r Ξ r ) i ] P 1 I e i + 10 [Ξ i Ξ ri ] P 1 D Te i + 11 [(D Ξ ) i (D r Ξ r ) i ] P 1 D Te i + at e1 01 [(T e1 ) i (T e1r ) i ] P 1 I e i ǫt e 11 [(T e1 ) i (T e1r ) i ] P 1 D T e i Ξ (x, 0) = 0. Ξ r t + ad r Ξ r ǫd r D r Ξ r = r 00 [Ξ ri Ξ i ] Pr 1 I r e i+ + r 01 [(D r Ξ r ) i (D r Ξ ) i ] Pr 1 I r e i+ + r 10 [Ξ ri Ξ i ] Pr 1 Dr T e i+ + r 11 [(D r Ξ r ) i (D r Ξ ) i ] Pr 1 Dr T e i+ + at e1r r 01 [(T e1r ) i (T e1 ) i ] Pr 1 I r e i ǫt er r 11 [(T e1r ) i (T e1 ) i ] Pr 1 Dr T e i Ξ r (x, 0) = 0. (59) Four distinct types of discretization errors appear in equation (59). The first two truncation error vectors: T e1 and T e, arise from errors associated with the approximation of the first and second derivative terms, respectivey. Like conventiona HOFDM operators, they are sedom uniformy accurate throughout the domain. Points near boundaries are commony discretized ess accuratey than those in the interior. Furthermore, the boundary stencis used in first and second difference operators are frequenty of different orders. The ast two error vectors resut from the interface penaty terms and have the forms: [(T e1 ) i (T e1 ) i ] P 1 I e i and [(T e1 ) i (T e1 ) i ] P 1 D T e i. Note that the accuracy of these vectors is infuenced both by the truncation error of the interface derivative operators, (T e1r ) i, and by the size of the penaty scaing terms. To assess the size of the scaing terms, first note that the differentiation operator and its transpose, [D and D T ] scae as O( x) 1. Further, since Q is unitess, and D = P 1 Q, the inverse norm operator P 1 scaes as O( x) 1. Thus, the ast two error vectors scae as O( x) r 1 and O( x) r, respectivey, where r is the oca accuracy of the interface derivative operator. In summary, if interface penaties satisfy p 1 then the error vectors in both domains have eading order truncation terms that scae as T e1 = [O( x r1 ),,O( x r1 ), O( x p ),, O( x p ), O( x r1 ),, O( x r1 ) ] T T e = [O( x r ),,O( x r ), O( x p ),, O( x p ), O( x r ),, O( x r ) ] T (T e1 ) i P 1 I T e i = [O( x r1 1 ),, O( x r1 1 ), 0,, 0, O( x r1 1 ),, O( x r1 1 )] T (T e1 ) i P 1 D T e i = [O( x r1 ),, O( x r1 ), 0,, 0, O( x r1 ),, O( x r1 )] T where the r1 and r exponents denote the boundary stenci order of accuracy in the first- and second-derivative operators, respectivey. (Here, we assume the interior stencis to be the same order of accuracy for both the first- and second-derivative operators.) 14 (60)

15 An error equation derived for the genera penaty technique presented in equation (11) woud invove consideraby more error terms. For exampe, extending equation (59) to the case p woud produce the foowing additiona terms 0 (T e ) i P 1 I T e i = O( x r 1 ) 1 (T e ) i P 1 D 1 T ei = O( x r ) (T e ) i P 1 D T ei = O( x r 3 ) (61) 1 (T e1 ) i P 1 D T ei = O( x r1 3 ) 0 P 1 D T ei = O( x p 3 ) as we as simiar expressions for r 0, r 1, r, r 1, r 0. Loca ower order error terms do not necessariy decrease the goba forma accuracy. The impact of ower-order terms on goba discretization accuracy is a ong-standing area of research that dates back to the pioneering work of Gustafsson [14]. There it was estabished that hyperboic operators coud accommodate oca terms one order ower than design order, whie sti maintaining forma accuracy. Recent work of Svärd and Nordström [34] has extended this resut to incude the impact of ow-order stencis for paraboic probems. Their work on goba accuracy is encapsuated in the foowing theorem. Theorem 8 If (1) is a pointwise stabe discretization of the continuous probem (9) for x x 0, then with interior operators D and DD satisfying discretization orders of p r1, r p at the boundary, and interface penaty terms satisfying discretization orders of p r1 p, then the goba order of accuracy of approximation (1) is p. Proof : See Svärd and Nordström [34], Theorems (.6) and (.8), pp for the origina theorems and proofs. Now consider the discretization of equation (9) using a reaistic combination of operators to determine the effects of each error term on goba accuracy. Assume that the derivative operator D is defined as in equation (60) with r1 = p 1. Furthermore, assume that the second derivative operator is formed as the action of two first derivative operators DD so that r = p. Finay, assume that the derivative used in the penaty (obtained from the derivative operator D) is O( x p 1 ), and is used in a penaty terms where necessary. In this scenario, the derivative operators D and DD maintain the design accuracy p of the method, as we as do the penaty terms producing errors of the form (T e1 ) i P 1. The remaining penaty terms producing errors of the form D T e i decrease the forma accuracy to p 1, at east in principe. Remark. Adding penaty terms of the form (T e1 ) i P 1 D T e i is i advised on a theoretica basis, athough experiments presented ater show that these estimates are not aways sharp. Furthermore, these experiments wi show these terms to be computationay impractica. Remark. Assume that j th -derivative operator D j, j p is r j -order accurate at the interface, and has interface truncation error T ej ) i. Interface penaties using these derivative operators produce terms in the error equation of the form jk (T ej ) i P 1 D kt e i = O( x rj (k+1) ) Remark. The reduction in accuracy is more severe for a hyperboic equation [e.g. ǫ = 0 in equation (9)]. Specificay, the boundary / penaty discretization orders required to achieve forma accuracy satisfies p 1 r1, r p. Thus, penaties of the form (T e1 ) i P 1 e i and (T e1 ) i P 1 D T e i decrease the forma accuracy to p 1 and p, respectivey. 4 Identification of schemes Specific vaues are assigned for the eighteen parameters used in the prima [equation (1)] and fux [equation (40)] formuations. Three popuar schemes are identified, incuding two DGFEM and one strong form HOFDM. Each is shown to satisfy their respective stabiity constraints. A three scheme ony penaize the soution and first derivative, so the parameters 0, 1,, 1, 0 and r 0, r 1, r, r 1, r 0 are set to zero. 15

16 4.1 Carpenter, Nordström, Gottieb [8] We begin with the scheme proposed by Carpenter et. a (CNG) [8] which is a specia case of the prima formuation presented in equation (1). This scheme was originay derived in strong form in the context of HOFDM. This scheme [See reference [8], equation (7).] can be reproduced by assigning the foowing specific vaues to the penaty parameters; 10 = 11 = r 10 = r 11 = 0 ; 00 = σ 1 ; 01 = ǫσ ; r 00 = σ 3 ; r 01 = ǫσ 4 ; (6) Substituting (6) into equation (9) yieds the conditions σ 1 = σ 3 + a ; [σ = σ 4 1] (63) 0 [(α + β)ǫ] ; [ǫ( α + β)] [(α + β)ǫ] ; σ 1 a ǫ[σ 4β + σ 4 4α ] (64) Further, the stabiity conditions obtained in equation (63) and (64) are identica to those found in the origina CNG work [e.g. equation (8) in reference [8]]. 4. Baumann, Oden [4] The scheme proposed by Baumann and Oden [4] is a specia case of the prima formuation presented in equation (1). This scheme was originay deveoped in the weak form in the context of DGFEM. Shu presents in reference [30], an iustrative comparison between this scheme and the LDG scheme for the stricty paraboic case. Therein the Baumann-Oden scheme is presented in equations (5.1) and (5.) and can be reproduced from equation (1) with the vaues 00 = r 00 = 11 = r 11 = 0 ; 01 = ǫ ; 10 = ǫ ; r 01 = + ǫ ; r 10 = + ǫ (65) Taking the iberty to extend Shu s definition to incude the possibiity of convection terms, we define the Baumann- Oden scheme by assigning the foowing vaues to the coefficients in equation (1): 00 = r 00 + a; 11 = r 11 = 0 ; 01 = +(β ǫ ); 10 = (β + ǫ ); r 01 = +(β + ǫ ); r 10 = (β ǫ ) (66) Substituting (66) into equation (9) yieds the conditions 0 [(α + β)ǫ] ; [ǫ( α + β)] [(α + β)ǫ] ; 00 a (67) 4.3 Loca Discontinuous Gaerkin [11] The scheme proposed by Cockburn and Shu [11] is specia case of the fux formuation presented in equation (40). Once again, Shu presents in reference [30], the LDG scheme written stricty for the paraboic case. Therein the LDG scheme is presented in equations (4.1), (4.), and (4.3) and can be reproduced from equation (40) with the vaues L 00 = R 00 = L 11 = R 11 = 0 ; L 01 = ǫ ; L 10 = ǫ ; R 01 = + ǫ ; R 10 = + ǫ (68) Again, taking the iberty to extend Shu s definition to incude the possibiity of convection terms, we define the coocation LDG scheme by assigning the foowing vaues to the coefficients in equation (40): L 00 = R 00 + a; L 11 = R 11 = 0 ; L 01 = +(β ǫ ); L 10 = (β + ǫ ); R 01 = +(β + ǫ ); R 10 = (β ǫ ) (69) Substituting (69) into equation (9) yieds the conditions 0 [(α + β)ǫ] ; [ǫ( α + β)] [(α + β)ǫ] ; L 00 a (70) 16

17 5 Numerica experiments 5.1 Test Probems The One-Way Wave Equation The inear one-way wave equation is used to study the stabiity and accuracy of the new formuations in the absence of diffusion terms. The functiona form is with the exact soution The initia and boundary data coincide with the exact soution for a time. U t + au x = 0 x 1, 0 t 0.1 (71) U(x, t) = cos[ π (x t)] (7) The one-way wave equation is used to estabish a baseine accuracy for each method, before moving on to the paraboic equation. This step is essentia to estabish the effects of the newy added terms that penaize the diffusive terms. Specificay, the new additiona terms coud destroy the accuracy of the advection term, not just diffusion. Comparing the two resuts aows us to determine the source of error. This study of the first order one-way wave equation raises a subte point regarding imposition of discrete interface conditions; specificay, that derivative fuxes can be penaized between eements, despite the fact that weposedness requires ony one interface condition. Thus, the diffusive interface penaty can be thought of as an aternative discrete stenci, one that weaky enforces first derivative continuity across the interface. An a priori stabiity estimate and accurate interface data guarantee the vaidity of these penaties Linear Burgers Equations The inear Burgers equation tests a combination of the advection and diffusion terms. Five distinct vaues of the diffusion parameter ǫ are used to deduce the effects of the penaty as the probem becomes diffusion dominated. The functiona form is U t + a U x = ǫu xx x 1, 0 t 0.1 (73) with the exact soution U(x, t) = exp[ ǫ b t] cos[ π (x t)] (74) with initia and boundary data that coincides with that of the exact soution. The equation parameters used are a = and b = π in a cases. Five vaues of the parameter ǫ are used: ǫ = 1, 5, , and Test Schemes Two finite difference and four spectra operators are tested using the inear wave and inear Burgers equation. The finite difference operators are the 4 th and 6 th order centra difference operators using SBP cosures at the boundaries. A finite difference simuations utiize a uniform grid within each eement, athough the eement size is aowed to vary. Bock-norm boundary cosures are used that are 3 rd - and 5 th -order accurate [6], respectivey. The boundary cosure formuas are those reported in reference [6]. The spectra operators are conventiona coocation operators defined using the Gauss-Lobatto-Chebyshev noda points. Poynomia orders in the range p 5 are used (corresponding to eements having three to six noda points). The penaties are impemented such that the scheme is equivaent to a Legendre coocation scheme (see Carpenter and Gottieb [7]). The time advancement scheme used in a cases is the five-stage fourth-order Runge-Kutta (RK) scheme [5]. The timestep is chosen such that the tempora error estimate is independent of further timestep reduction. The magnitude of the tempora error is approximatey 10 1, and is obtained by comparing the fourth-order soution with an embedded third-order soution. Thus, the spatia error is the dominant component of error in the simuations. 17

18 x j 3/ Ij 1 x j 1/ x j+1/ Ij+1 x j+3/ x j+5/ Ij Ij+ Figure : Schematic depicting two periods of the recursive eement pattern used in the nonuniform grid accuracy studies. 5.3 Test Matrix An extensive test matrix is used to distinguish different properties of the new formuations. The matrix incudes 1) four geometry definitions, and ) mutipe vaues of three distinct interface penaty parameters. The geometric definitions incuded a permutations of uniform and nonuniform eement distributions on both periodic and nonperiodic intervas, specificay; 1) uniform-periodic, ) nonuniform-periodic, 3) uniform-nonperiodic, and 4) nonuniform-nonperiodic. The nonuniform grids are generated using eements that aternate in size with a ratio of : 1. Figure () shows a schematic depicting two periods of the : 1 nonuniform grid used in the study. An a priori study is performed, comparing the accuracy of the methods on each of the four different geometries. The resuts for these studies are avaiabe in reference [9]. The most chaenging test cases (hyperboic or paraboic) are those that were run on the nonuniform grid, with nonperiodic boundary conditions. Thus, the nonuniform grid with nonperiodic boundary conditions is used amost excusivey in this work. The interface penaties are parametrized using three independent variabes as (here written for the prima formuation), 00 = a (1 α); r 00 = a ( 1 α); α 0 (75) 01 = +(β ǫ ); 10 = (β + ǫ ); r 01 = +(β + ǫ ); r 10 = (β ǫ ) (76) 11 = r 11 = γ; γ 0 (77) The parameter α adjusts the contribution of the Dirichet penaty between the eft and right states. The specia vaue α = 0 admits no dissipation at the interface, thus producing a skew-symmetric matrix in the P-norm. The vaue α = 1 produces a fuy upwind fux. Two vaues of the parameter α are tested: α = 0 and α = 1. The parameters β and γ adjust the penaty on the derivative fuxes. For β the reationship between the vaues of 01, 10, r 01, and r 10, greaty simpifies the energy estimate by producing the conditions t 1 = t = 0 in the estimate. This combination of parameters adds no dissipation to the energy estimate. Note that the vaue of β can be chosen independent from ǫ, and can be nonzero for vaues of ǫ = 0. Two vaues of the parameter β are tested: β = 0 and β = 1 4. The parameter γ adjusts the contribution from the derivative fuxes and produces contributions to the energy estimate that are stricty dissipative. Four vaues of γ are tested: γ = 0, γ = 10 4, γ = 10 3 and γ = 10. A studies incude a comparison of the convergence rate of each scheme. A grid refinement exercise invoving approximatey 40 grid densities is performed for each set of parameters. The convergence rate is determined by fitting a east-squares curve through the error data. 5.4 Resuts Finite Difference: One-Way Wave and Linear Burgers Equation Figure (3) compares the convergence behavior of the 4 th - and 6 th -order finite difference schemes. The mode probem used is the inear one-way wave equation. Shown are the resuts obtained on a nonuniform grid with nonperiodic boundary conditions. Seven sets of interface parameters are compared, and in a cases, design order convergence is 18