Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

Transcription

1 Journal of Scientific Computing, Vol. 17, Nos. 1 4, December 00 ( 00) Local Discontinuous Galerkin Methos for Partial Differential Equations with Higher Orer Derivatives Jue Yan 1 an Chi-Wang Shu 1 Receive November 5, 001; accepte December 1, 001 In this paper we review the existing an evelop new local iscontinuous Galerkin methos for solving time epenent partial ifferential equations with higher orer erivatives in one an multiple space imensions. We review local iscontinuous Galerkin methos for convection iffusion equations involving secon erivatives an for KV type equations involving thir erivatives. We then evelop new local iscontinuous Galerkin methos for the time epenent bi-harmonic type equations involving fourth erivatives, an partial ifferential equations involving fifth erivatives. For these new methos we present correct interface numerical fluxes an prove L stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methos. Finally, we present new results on a post-processing technique, originally esigne for methos with goo negative-orer error estimates, on the local iscontinuous Galerkin methos applie to equations with higher erivatives. Numerical experiments show that this technique works as well for the new higher erivative cases, in effectively oubling the rate of convergence with negligible aitional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh. KEY WORDS: Discontinuous Galerkin metho; partial ifferential equations with higher erivatives; stability; error estimate; post-processing. 1. INTRODUCTION In this paper we review existing an evelop new local iscontinuous Galerkin methos for solving time epenent partial ifferential equations with higher orer erivatives in one an multiple space imensions. We consier a sequence of such partial ifferential equations with increasingly higher orer erivatives. A hyperbolic conservation law U t + C f i (U) xi =0 (1.1) i=1 1 Division of Applie Mathematics, Brown University, 18 George Street, Provience, Rhoe Islan {yjue;shu}@cfm.brown.eu /0/ /0 00 Plenum Publishing Corporation

2 8 Yan an Shu is a partial ifferential equation with first erivatives. The convection iffusion equation U t + C i=1 f i (U) xi C i=1 C (a ij (U) U xj ) xi =0 (1.) j=1 where (a ij (U)) is a symmetric, semi-positive efinite matrix, is a partial ifferential equation with secon erivatives. The general KV type equation U t + C i=1 f i (U) xi + C 1 r i=1 i(u) C j=1 g ij (r i (U) xi ) xjx i =0 (1.3) is a partial ifferential equation with thir erivatives. The time epenent bi-harmonic equation U t + C i=1 f i (U) xi + C (a i (U xi )U xi x i ) xi x i =0 (1.4) i=1 is a partial ifferential equation with fourth erivatives, where the nonlinearity coul be more general but we just present (1.4) as an example. The following equation U t + C i=1 f i (U) xi + C g i (U xi x i ) xi x i x i =0 (1.5) i=1 is a partial ifferential equation with fifth erivatives, where again the nonlinearity coul be more general but we just present (1.5) as an example. Similar equations with sixth or higher erivatives coul also be presente. All these equations, an their time inepenent steay state counterparts, appear often in physical an engineering applications. In this paper we use capital letters U etc. to enote the solutions to the PDEs an lower case letters to enote the numerical solutions. The type of iscontinuous Galerkin methos we will iscuss in this paper, using a iscontinuous Galerkin finite element approximation in the spatial variables couple with explicit, nonlinearly stable high orer Runge Kutta time iscretization [19], were first evelope for the conservation laws (1.1) containing first erivatives by Cockburn et al. in a series of papers [8, 9, 6, 4, 10]. We will briefly review this metho in Section.1. For a etaile escription of the metho as well as its implementation an applications, we refer the reaers to the lecture notes [3], the survey paper [5], other papers in that Springer volume, an the review paper [1]. For equations containing higher orer spatial erivatives, iscontinuous Galerkin methos cannot be irectly applie. This is because the solution space, which consists of piecewise polynomials iscontinuous at the

3 Local Discontinuous Galerkin Methos for Partial Differential Equations 9 element interfaces, is not regular enough to hanle higher erivatives. This is a typical non-conforming case in finite elements. A naive an careless application of the iscontinuous Galerkin metho irectly to the heat equation containing secon erivatives coul yiel a metho which behaves nicely in the computation but is inconsistent with the original equation an has O(1) errors to the exact solution [18, 1]. The iea of local iscontinuous Galerkin methos for time epenent partial ifferential equations with higher erivatives is to rewrite the equation into a first orer system, then apply the iscontinuous Galerkin metho on the system. A key ingreient for the success of such methos is the correct esign of interface numerical fluxes. These fluxes must be esigne to guarantee stability an local solvability of all the auxiliary variables introuce to approximate the erivatives of the solution. The local solvability of all the auxiliary variables is why the metho is calle a local iscontinuous Galerkin metho in [11]. The first local iscontinuous Galerkin metho was evelope by Cockburn an Shu [11], for the convection iffusion equation (1.) containing secon erivatives. Their work was motivate by the successful numerical experiments of Bassi an Rebay [1] for the compressible Navier Stokes equations. Later, Yan an Shu [0] evelope a local iscontinuous Galerkin metho for the general KV type equation (1.3) containing thir erivatives. In both [11] an [0], suitable numerical fluxes at element interfaces were given, which le to provable nonlinear L stability of the methos as well as error estimates for the linear cases. Numerical examples were shown to illustrate the stability an accuracy of these methos. These results will be briefly reviewe in Sections. an.3, to motivate the new evelopment in later sections. In Section 3 of this paper we evelop new local iscontinuous Galerkin methos for the time epenent bi-harmonic type equation (1.4) involving fourth erivatives, an in Section 4 we o the same thing for the partial ifferential equation (1.5) involving fifth erivatives. Similar methos can be esigne for well pose partial ifferential equations involving even higher erivatives. As before, we give recipes for correct inter-element numerical fluxes which lea to provable nonlinear L stability of the methos. The stability result is inepenent of the coefficients in front of the higher orer erivatives, hence the methos are especially suitable for the so-calle convection ominate problems, i.e., those with small coefficients to the higher erivative terms an hence are ominate by the first erivative convection terms. Also, these methos are extremely local an hence efficient for parallel implementations an easy for h p aaptivity. We will provie preliminary numerical examples to verify the stability an accuracy of these methos. A post-processing technique was introuce in [7], which is a local convolution if the mesh is locally uniform an has been proven to be able to recover the solution to an accuracy of Dx k+1 rather than the usual Dx k+1 when piecewise polynomials of egree k are use, for linear

4 30 Yan an Shu conservation laws (1.1) (theoretically proven) an for linear convection iffusion equations (1.) (numerically observe). The post-processing is even useful in enhancing accuracies for some nonlinear cases in numerical experiments. The key ingreient of this post-processing technique is a negative-orer error estimate, thus if the problem is linear, the solution is sufficiently smooth, an the metho has a goo negative-orer error estimate, then the post-processing will achieve the orer of convergence of the negative-orer norm (which is usually bigger than the orer of convergence of the L -norm of the error in Galerkin methos). In Section 5 of this paper, we apply this post-processing technique to our new local iscontinuous Galerkin metho in [0] an in this paper, for the linear KV like equations (1.3), linear time epenent bi-harmonic equations (1.4), an the linear PDE (1.5) containing fifth erivatives. We observe the same accuracy enhancement capability for all these cases, where accuracy has been increase from Dx k+1 before post-processing to Dx k+1 or higher (in many cases Dx k+ ) after post-processing. This strongly suggests that the LDG methos we have esigne for these equations with higher erivatives have an orer of convergence of k+1 or higher in the negative-orer norm, when piecewise polynomials of egree k are use. The post-processing technique thus is fully taking avantage of this. To emonstrate that this accuracy enhancement can even be observe for some nonlinear problems, we also apply it to the local iscontinuous Galerkin metho in [0] for the nonlinear KV equations. As the post-processing step is applie only at the en of the computation an is applie only locally, the aitional computational cost is negligible. Concluing remarks summarizing results in this paper an inicating future work are inclue in Section 6.. REVIEW OF THE DISCONTINUOUS GALERKIN METHODS FOR PDES WITH FIRST, SECOND, AND THIRD DERIVATIVES In this section we review the essential points of the iscontinuous Galerkin methos for the conservation laws (1.1) with first erivatives, the local iscontinuous Galerkin methos for the convection iffusion equations (1.) with secon erivatives an the KV type equations (1.3) with thir erivatives. For simplicity of presentation, we will use one imensional cases to present the methos. However, we will inicate which results are also vali for the general multi-imensional cases..1. Conservation Laws The one imensional version of the conservation laws (1.1) has the form U t +f(u) x =0 (.1)

5 Local Discontinuous Galerkin Methos for Partial Differential Equations 31 First let s introuce some notations. The computational mesh is given by I j =[x j 1,x j+ 1 ], for j=1,..., N, with the center of the cell enote by x j = 1 (x j 1 +x j+ 1 ) an the size of each cell by Dx j=x j+ 1 x j 1. We will enote Dx=max j Dx j. If we multiply (.1) by an arbitrary test function V(x), integrate over the interval I j, an integrate by parts, we get U t Vx f(u) V x x+f(u(x j+ 1, t)) V(x j+ 1 ) f(u(x j 1, t)) V(x j 1 )=0 (.) This is the starting point for esigning the iscontinuous Galerkin metho. The semi-iscrete version of the iscontinuous Galerkin metho [9, 6, 4, 10] can be escribe as follows: we replace both the solution U an the test function V by piecewise polynomials of egree at most k, an enote them by u an v. That is, u, v V Dx where V Dx ={v : v is a polynomial of egree at most k for x I j, j=1,..., N} (.3) With this choice, there is an ambiguity in (.) in the last two terms involving the bounary values at x j± 1, as both the solution u an the test function v are iscontinuous exactly at these bounary points. A crucial ingreient for the success of iscontinuous Galerkin metho for the conservation laws is the correct choice of the numerical fluxes to overcome (or we coul say it in a positive way, to utilize) this ambiguity of iscontinuity at the element interfaces. The iea is to treat these terms by an upwining mechanism (information from characteristics), borrowe from the successful high resolution finite volume schemes. Thus f(u) at the interface x j+ 1 for each j is given by a single value monotone numerical flux fˆj+ 1 =fˆ(u j+ 1,u+ j+ 1 ) (.4) which epens both on the left limit u j+ an on the right limit u + 1 j+ of the 1 iscontinuous numerical solution at the interface x j+ 1. Here monotone flux means that the function fˆ is a non-ecreasing function of its first argument an a non-increasing function of its secon argument. It is also assume to be at least Lipschitz continuous with respect to each argument an to be consistent with the physical flux f(u) in the sense that fˆ(u, u)=f(u). If (.1) is a system rather than a scalar equation, then the monotone flux is replace by an exact or approximate Riemann solver, see, e.g., [16] for etails. On the other han, the test function v at the interfaces x j± 1 is taken from insie the cell I j, namely v j+ an v + 1 j respectively. The final 1 semi-iscrete scheme u t f(u) v x x+fˆj+ 1 vx FI v j+ 1 fˆj 1 v + j =0 (.5) 1 j

6 3 Yan an Shu is then iscretize by a nonlinearly stable high orer Runge Kutta time iscretizations [19]. Nonlinear TVB limiters [17] may be use if the solution contains strong iscontinuities. The schemes thus obtaine have the following attractive properties, for the general multi-imensional case (1.1) with arbitrary triangulations: 1. It can be easily esigne for any orer of accuracy. In fact, the orer of accuracy can be locally etermine in each cell, thus allowing for efficient p aaptivity.. It can be use on arbitrary triangulations, even those with hanging noes, thus allowing for efficient h aaptivity. 3. It is extremely local in ata communications. The evolution of the solution in each cell nees to communicate only with the immeiate neighbors, regarless of the orer of accuracy, thus allowing for efficient parallel implementations. See, e.g., []. These schemes also have the following provable theoretical properties, all of these are vali also for the multi-imensional case (1.1) with arbitrary triangulations: 1. The semi-iscrete scheme (.5), an certain time iscretization of it, such as implicit backwar Euler an Crank Nicholson, have excellent nonlinear stability properties. One can prove a strong L stability an a cell entropy inequality for the square entropy, for the general nonlinear scalar case (1.1), for any orers of accuracy on arbitrary triangulations in any space imension, without the nee for nonlinear limiters [14]. Notice that these stability results are vali even when the solution contains iscontinuities such as shocks.. For linear problems with smooth solutions, these methos using piecewise polynomials of egree k have a provable error estimate of orer Dx k+1 in L [15]. In effect, for most triangulations one coul observe (an prove in many cases) convergence of the orer Dx k+1 in both L an L. norms, for both linear an nonlinear problems. 3. When nonlinear TVB limiters [17, 9, 4] are use, the methos can be proven stable in the total variation norm for scalar one imensional nonlinear problems (.1), an stable in the L. norm for scalar multi-imensional nonlinear problems (1.1)... Convection Diffusion Equations The one imensional version of the convection iffusion equation (1.) has the form U t +f(u) x (a(u) U x ) x =0 (.6) where a(u) \ 0.

7 Local Discontinuous Galerkin Methos for Partial Differential Equations 33 The semi-iscrete version of the local iscontinuous Galerkin metho for solving (.6) [11] approximates the following lower orer system U t +f(u) x (b(u) Q) x =0, Q B(U) x =0 (.7) where b(u)=`a(u), an B(U)=> U b(u) U. We can then formally use the same iscontinuous Galerkin metho for the convection equation to solve (.7), resulting in the following scheme: fin u, q V Dx such that, for all test functions v, w V Dx, u t vx F I j (f(u) b(u) q) v x x+(fˆj+ 1 bˆ j+ 1 qˆ j+ 1 )v j+ 1 (fˆj 1 bˆ j 1 qˆ j 1 )v + j 1 =0 qw x+ B(u) w x x Bˆ j+ 1 w j+ 1 +Bˆ j 1 w+ j 1 =0 (.8) Again, a crucial ingreient for the metho to be stable is the correct choice of the numerical fluxes (the hats ). However, there is no longer a upwining mechanism or characteristics to guie the esign of these fluxes. In [11], criteria are given for these fluxes to guarantee stability an convergence. The best choice is to use the alternating principle in esigning the fluxes: bˆ= B(u+ ) B(u ) u + u ; qˆ=q ; Bˆ =B(u + ) (.9) an fˆ is chosen as before in (.4). Notice that we i not write the subscript j+ 1 for the fluxes as they are all evaluate at this interface point. The alternating principle refers to the alternating choices of qˆ an Bˆ : if the left value is chosen for the former then the right value is chosen for the latter, as in (.9). One coul also choose qˆ=q + ; Bˆ =B(u ) with all the other fluxes unchange. These choices of fluxes guarantee stability an optimal convergence. We remark that the appearance of the auxiliary variable q is superficial: when a local basis is chosen in cell I j then q is eliminate by using the secon equation in (.8) an solving a small linear system if the local basis is not orthogonal. The actual scheme for u takes a form similar to that for convection alone. This is a big avantage of the scheme over the traitional mixe methos, whose auxiliary variable is usually genuinely global. This is the reason that the scheme is terme local iscontinuous Galerkin metho in [11]. We also remark that the choice of fluxes in (.9) by the

8 34 Yan an Shu alternating principle yiels the most compact stencil for the scheme for u after the auxiliary variable q is locally eliminate. The schemes thus esigne for the one imensional equation (.6), or in fact for the most general multi imensional nonlinear convection iffusion equations (1.), which is nonlinear both in the first erivative convection part an in the secon erivation iffusion part, retain all of the three attractive properties liste above for the metho use on convection equations. They also have the following provable theoretical properties, all of these are vali for the multi-imensional case (1.) with arbitrary triangulations [11]: 1. The semi-iscrete scheme (.8), an certain time iscretization of it, such as implicit backwar Euler an Crank Nicholson, have excellent nonlinear stability properties. One can prove a strong L stability for the general nonlinear scalar case (1.), for any orers of accuracy on arbitrary triangulations in any space imension. This stability is inepenent of the size of the iffusion terms an hence is also vali in the limit when the iffusion coefficient goes to zero.. For linear problems with smooth solutions, these methos using piecewise polynomials of egree k have a provable error estimate of orer Dx k in L. In effect, for most triangulations one coul observe (an prove in many cases) convergence of the orer Dx k+1 in both L an L. norms, for both linear an nonlinear problems. The same error estimate can be obtaine also for the auxiliary variable q, which approximates the erivative U x, even if q can be locally eliminate in actual calculation..3. KV Like Equations The one imensional version of the KV like equation (1.3) has the form U t +f(u) x +(rœ(u) g(r(u) x ) x ) x =0 (.10) where f(u), r(u) an g(u) are arbitrary functions. The semi-iscrete version of the local iscontinuous Galerkin metho for solving (.10) [0] approximates the following lower orer system U t +(f(u)+rœ(u) P) x =0, P g(q) x =0, Q r(u) x =0 (.11) We can then formally use the same iscontinuous Galerkin metho for the convection equation to solve (.10), resulting in the following scheme: fin u, p, q V Dx such that, for all test functions v, w, z V Dx,

9 Local Discontinuous Galerkin Methos for Partial Differential Equations 35 u t (f(u)+rœ(u) p) v x x+(fˆ+rœ 5 pˆ)j+ 1 vx FI v (fˆ+rœ 5 j+ 1 pˆ)j 1 v + =0, j 1 j pw x+ g(q) w x x ĝ j+ 1 w j+ 1 +ĝ j 1 w+ j 1 =0, qz r(u) z x x rˆj+ 1 x+fi z j+ 1 +rˆj 1z+ =0 j 1 j (.1) Again, a crucial ingreient for the metho to be stable is the correct choice of the numerical fluxes (the hats ). It is foun out in [0] that one can take the following simple choice of fluxes to guarantee stability an convergence: fˆ=fˆ(u,u + ), rœ 5 r(u + ) r(u ) =, pˆ=p +, ĝ=ĝ(q,q + ), rˆ=r(u ) u + u (.13) where fˆ(u,u + ) is a monotone flux for f(u), an ĝ(q,q + ) is a monotone flux for g(q). In fact, the crucial part is still the alternating principle to take pˆ an rˆ from opposite sies. Thus fˆ=fˆ(u,u + ), rœ 5 r(u + ) r(u ) =, pˆ=p, ĝ=ĝ(q,q + ), rˆ=r(u + ) u + u woul also work. Again, the appearance of the auxiliary variables p an q is superficial: when a local basis is chosen in cell I j then both of them can be eliminate by using the secon an thir equations in (.1) an solving two small linear systems if the local basis is not orthogonal. The actual scheme for u takes a form similar to that for convection alone. We also remark that the choice of fluxes in (.13) by the alternating principle yiels a compact stencil for the scheme for u after the auxiliary variables p an q are locally eliminate. The schemes thus esigne for the KV like equation (.10), or in fact for the most general multi imensional nonlinear KV like equations (1.3), which is nonlinear in all the erivatives, retain all of the three attractive properties liste above for the metho use on convection equations. They also have the following provable theoretical properties, all of these are vali for the multi-imensional case (1.3) with arbitrary triangulations [0]: 1. The semi-iscrete scheme (.1), an certain time iscretization of it, such as implicit backwar Euler an Crank Nicholson, have excellent nonlinear stability properties. One can prove a strong L stability an a cell entropy inequality for the square entropy, for the general nonlinear scalar case (1.3), for any orers of accuracy

10 36 Yan an Shu on arbitrary triangulations in any space imension, without the nee for nonlinear limiters [0]. Notice that these stability results are vali even in the limit when the coefficients of the ispersive thir erivative terms ten to zero.. For one imensional linear problems with smooth solutions, these methos using piecewise polynomials of egree k have a provable error estimate of orer Dx k+1 in L [0]. In numerical experiments one observes convergence of the orer Dx k+1 in both L an L. norms for both one an multiple imensional linear an nonlinear cases. 3. A LOCAL DISCONTINUOUS GALERKIN METHOD FOR THE BI-HARMONIC TYPE EQUATIONS In this section, we present an analyze a LDG metho for the bi-harmonic type equation (1.4). We will concentrate on the one imensional case with an initial conition U t +f(u) x +(a(u x )U xx ) xx =0, 0 [ x [ 1 (3.1) U(x, 0)=U 0 (x), 0 [ x [ 1 (3.) an perioic bounary conitions. Here a(u x ) \ 0. The assumption of perioic bounary conitions is for simplicity only an is not essential: the metho can be easily esigne for non-perioic bounary conitions. The generalization to the multiple imensional case (1.4) is straightforwar, following the lines in [11] an [0]. To efine the LDG metho, we first introuce the new variables R=U x, Q=B(R) x, P=(b(R) Q) x (3.3) where b(r)=`a(r) an B(R)=> R b(r) R, an rewrite the Eq. (3.1) as a first orer system: U t +(f(u)+p) x =0, P (b(r) Q) x =0, Q B(R) x =0, R U x =0 (3.4) The LDG metho is obtaine by iscretizing the above system with the iscontinuous Galerkin metho. This is achieve by multiplying the four equations in (3.4) by four test functions v, w, z, s respectively, integrate over the interval I j, an integrate by parts. We nee to pay special attention to the bounary terms resulting from the proceure of integration by parts. Thus we seek piecewise polynomial solutions u, p, q, r V Dx, where V Dx is

11 Local Discontinuous Galerkin Methos for Partial Differential Equations 37 efine in (.3), such that for all test functions v, w, z, s V Dx we have, for 1 [ j [ N, u t vx (f(u)+p) v x x+(fˆ+pˆ) j+ 1 v j+ 1 (fˆ+pˆ) j 1 v + j 1 =0, pw x+ b(r) qw x x (bˆqˆ) j+ 1 w j+ 1 +(bˆqˆ) j 1 w + j 1 =0, qz x+ B(r) z x x Bˆ j+ 1 z j+ 1 +Bˆ j 1 z+ j 1 =0, (3.5) rs x+ us x x û j+ 1 s j+ 1 +û j 1 s+ j 1 =0 The only ambiguity in the algorithm (3.5) now is the efinition of the numerical fluxes (the hats ), which shoul be esigne to ensure stability. It turns out that we can take the simple choices (we omit the subscripts j± 1 in the efinition of the fluxes as all quantities are evaluate at the interfaces x j± 1 ) fˆ=fˆ(u,u + ), pˆ=p, bˆ= B(r+ ) B(r ), qˆ=q +, Bˆ =B(r ), û=u + r + r (3.6) where fˆ(u,u + ) is a monotone flux for f(u) in (.4). Here again, the alternating principle in esigning the fluxes relate to the fourth erivatives is at play: the fluxes for p, q, B(r) an u are alternatively taken from left an right. We coul thus also take fˆ=fˆ(u,u + ), pˆ=p +, bˆ= B(r+ ) B(r ), qˆ=q, Bˆ =B(r + ), û=u r + r (3.7) an stability is still vali. We remark again that the appearance of the auxiliary variables p, q an r is superficial: when a local basis is chosen in cell I j then all of them can be eliminate by using the secon, thir, an fourth equations in (3.5) an solving three small linear systems if the local basis is not orthogonal. The actual scheme for u takes a form similar to that for convection alone. We have the following L stability result for the scheme (3.5) (3.6). This is similar to the stability result in Proposition.1 in [11] for the secon orer convection iffusion equations. Proposition 3.1 (L Stability). The solution of the scheme (3.5) (3.6) satisfies t F u (x, t) 1 x+f q (x, t) x [ 0 (3.8) 0

12 38 Yan an Shu Proof. notation We sum up the four equalities in (3.5) an introuce the B j (u, p, q, r; v, w, z, s) = u t vx F I j (f(u)+p) v x x+(fˆ+pˆ) j+ 1 v j+ 1 (fˆ+pˆ) j 1 v + j 1 +F I j +(bˆqˆ) j 1 w + j 1 +F I j pw x+f I j b(r) qw x x (bˆqˆ) j+ 1 w j+ 1 qz x+f I j B(r) z x x Bˆ j+ 1 z j+ 1 +Bˆ j 1 z+ j 1 +F I j rs x+f I j us x x û j+ 1 s j+ 1 +û j 1 s+ j 1 (3.9) Clearly, the solutions u, p, q, r of the scheme (3.5) (3.6) satisfy for all v, w, z, s V Dx. We then take B j (u, p, q, r; v, w, z, s)=0 (3.10) v=u, w=r, z=q, s= p to obtain, after some algebraic manipulations, 0=B j (u, p, q, r; u, r, q, p) = t F I j 1 u (x, t) x+f q (x, t) x+(ĥ j+ 1 I Ĥ j 1 )+G j 1 (3.11) j with the numerical entropy flux Ĥ efine by Ĥ= F(u ) p u +B(r )q +(fˆ+pˆ) u (bˆqˆ) r Bˆ q +ûp an the extra term G given by G=[F(u)+pu B(r) q] (fˆ+pˆ)[u]+bˆqˆ[r]+bˆ [q] û[p] Here F(u)=> u f(u) u, an [v]=v + v enotes the jump of v. Notice that we have roppe the subscripts about the location j 1 or j+1 as all these quantities are efine at a single interface an epen only on the left an right values at that interface. Now all we nee to o is to verify G \ 0. To this en, we notice that, with the efinition (3.6) of the numerical fluxes an with simple algebraic manipulations, we easily obtain [pu B(r) q] pˆ[u]+bˆqˆ[r]+bˆ [q] û[p]=0

13 Local Discontinuous Galerkin Methos for Partial Differential Equations 39 an hence G=[F(u)] fˆ[u]=f u+ u (f(s) fˆ(u,u + )) s \ 0 because of the monotonicity of the flux fˆ. Summing up (3.11) over j woul now give the esire L stability (3.8). i For a preliminary numerical example of the methos evelope in this section, see Example 5.3 in Section A LOCAL DISCONTINUOUS GALERKIN METHOD FOR EQUATIONS WITH FIFTH DERIVATIVES In this section, we present an analyze a LDG metho for the Eq. (1.5) with fifth erivatives. We will concentrate on the one imensional case with an initial conition U t +f(u) x +g(u xx ) xxx =0, 0 [ x [ 1 (4.1) U(x, 0)=U 0 (x), 0 [ x [ 1 (4.) an perioic bounary conitions. Here g(u xx ) is an arbitrary function of U xx. The form of the nonlinearity is not general an is chosen simply as an example. The assumption of perioic bounary conitions is also for simplicity only an is not essential: the metho can be easily esigne for non-perioic bounary conitions. The generalization to the multiple imensional case (1.5) is straightforwar, following the lines in [11] an [0]. To efine the LDG metho, we first introuce the new variables S=U x, R=S x, Q=g(R) x, P=Q x (4.3) an rewrite the Eq. (4.1) as a first orer system: U t +(f(u)+p) x =0, P Q x =0, Q g(r) x =0, R S x =0, S U x =0 (4.4) The LDG metho is obtaine by iscretizing the above system with the iscontinuous Galerkin metho. This is achieve by multiplying the five equations in (4.4) by five test functions v, w, y, z, respectively, integrate over the interval I j, an integrate by parts. We nee to pay special attention to the bounary terms resulting from the proceure of integration by parts. Thus we seek piecewise polynomial solutions u, p, q, r, s V Dx, where

14 40 Yan an Shu V Dx is efine in (.3), such that for all test functions v, w, y, z, V Dx we have, for 1 [ j [ N, u t vx (f(u)+p) v x x+(fˆ+pˆ) j+ 1 v j+ 1 (fˆ+pˆ) j 1 v + j 1 =0, pw x+ qw x x qˆ j+ 1 w j+ 1 +qˆ j 1 w+ j 1 =0, qy x+ g(r) y x x ĝ j+ 1 y j+ 1 +ĝ j 1 y+ j 1 =0, (4.5) rz x+ sz x x ŝ j+ 1 z j+ 1 +ŝ j 1 z+ j 1 =0, s x+ u x x û j+ 1 j+ 1 +û j 1 + j 1 =0 The only ambiguity in the algorithm (4.5) now is the efinition of the numerical fluxes (the hats ), which shoul be esigne to ensure stability. It turns out that we can again take the simple choices fˆ=fˆ(u,u + ), pˆ=p, qˆ=q +, ĝ=ĝ(r,r + ), ŝ=s, û=u + (4.6) where fˆ(u,u + ) is a monotone flux for f(u) in (.4), an ĝ(r,r + ) is a monotone flux for g(r). Here again, the alternating principle in esigning the fluxes relate to the fifth erivatives is at play: the fluxes for p, q, s an u are alternatively taken from left an right. We coul thus also take fˆ=fˆ(u,u + ), pˆ=p +, qˆ=q, ĝ=ĝ(r,r + ), ŝ=s +, û=u (4.7) an stability is still vali. As before, the appearance of the auxiliary variables p, q, r an s is superficial: when a local basis is chosen in cell I j then all of them can be eliminate by using the secon, thir, fourth an fifth equations in (4.5) an solving four small linear systems if the local basis is not orthogonal. The actual scheme for u takes a form similar to that for convection alone. We have the following cell entropy inequality for the square entropy, which implies L stability, when summe up over j, for the scheme (4.5) (4.6). This is similar to the cell entropy inequality in [14] for conservation laws an the one in [0] for KV type equations involving thir erivatives.

15 Local Discontinuous Galerkin Methos for Partial Differential Equations 41 Proposition 4.1 (Cell Entropy Inequality). (4.5) (4.6) satisfies t F I j for some numerical entropy flux Ĥ j+ 1. The solution of the scheme 1 u (x, t) x+(ĥj+ 1 Ĥ j 1 ) [ 0 (4.8) Proof. notation We sum up the five equalities in (4.5) an introuce the B j (u, p, q, r, s; v, w, y, z, ) = u t vx F I j (f(u)+p) v x x+(fˆ+pˆ) j+ 1 v j+ 1 (fˆ+pˆ) j 1 v + j 1 +F I j +qˆ j 1 w + j 1 +F I j pw x+f I j qw x x qˆ j+ 1 w j+ 1 qy x+f I j g(r) y x x ĝ j+ 1 y j+ 1 +ĝ j 1 y + j 1 +F I j rz x+f I j sz x x ŝ j+ 1 z j+ 1 +ŝ j 1 z+ j 1 + s x+f I j u x x û j+ 1 j+ 1 +û j 1 + j 1 (4.9) Clearly, the solutions u, p, q, r an s of the scheme (4.5) (4.6) satisfy for all v, w, y, z, V Dx. We then take B j (u, p, q, r, s; v, w, y, z, )=0 (4.10) v=u, w=s, y= r, z=q, = p to obtain, after some algebraic manipulations, 0=B j (u, p, q, r, s; u, s, r, q, p) = t F I j with the numerical entropy flux Ĥ efine by 1 u (x, t) x+(ĥj+ 1 Ĥ j 1 )+G j 1 (4.11) Ĥ= F(u ) p u +q s G(r )+(fˆ+pˆ) u qˆs +ĝr ŝq +ûp

16 4 Yan an Shu an the extra term G given by G=[F(u)+pu qs+g(r)] (fˆ+pˆ)[u]+qˆ[s] ĝ[r]+ŝ[q] û[p] Here F(u)=> u f(u) u an G(r)=> r g(r) r, an [v]=v + v enotes the jump of v. Now all we nee to o is to verify G \ 0. To this en, we notice that, with the efinition (4.6) of the numerical fluxes an with simple algebraic manipulations, we easily obtain an hence [pu qs] pˆ[u]+qˆ[s]+ŝ[q] û[p]=0 G=[F(u)] fˆ[u] [G(r)]+ĝ[r] =F u+ \ 0 u (f(t) fˆ(u,u + )) t F r + r (g(t) ĝ(r,r + )) t where the last inequality follows from the monotonicity of the fluxes fˆ an ĝ. This finishes the proof. i For a preliminary numerical example of the methos evelope in this section, see Example 5.4 in Section POST-PROCESSING AND NUMERICAL RESULTS In [7], an efficient local post-processing technique has been evelope, which is applie only at the en of the calculation to the numerical solution at t=t, an involves only a local linear operation (convolution) using the information from a fixe (inepenent of the mesh size Dx) number of neighboring cells. The escription of this post-processing proceure is omitte here because of space limitation. Please see [7] for etails. It is proven in [7] that this post processing will improve the orer of accuracy from Dx k+1 to Dx k+1 when piecewise polynomials of egree k are use, for linear conservation laws (1.1) with a uniform (or locally uniform) mesh with smooth solutions. Although not proven theoretically, it is also verifie numerically in [7] that the same enhancement of accuracy can be observe for linear convection iffusion equations (1.). In fact, the crucial ingreient this post-processing technique epens on is a negative-orer error estimate, which is usually higher than the orer of convergence of the L -norm of the error, for Galerkin type methos. If the problem is linear, the solution is sufficiently smooth, then the post-processing will achieve the orer of convergence of the negative-orer norm. In this section, we perform numerical experiments of this post-processing techniques when the local iscontinuous Galerkin metho is applie

17 Local Discontinuous Galerkin Methos for Partial Differential Equations 43 to the linear an nonlinear KV equations (1.3), an linear versions of (1.4) an (1.5) containing fourth an fifth erivatives. We observe the same accuracy enhancement. This strongly suggests that the LDG methos we have esigne for these equations with higher erivatives have an orer of convergence of k+1 or higher in the negative-orer norm, when piecewise polynomials of egree k are use. The post-processing technique thus is fully taking avantage of this. Time iscretization is by the thir an fourth orer implicit Runge Kutta methos in [13]. An efficient time marching of these local iscontinuous Galerkin methos without sacrificing its local property an parallel flexibility is left for future work. We have chosen Dt suitably small so that spatial errors ominate in the numerical results. Example 5.1. We solve the linear KV equation (1.3) in 1D: U t +U xxx =0 (5.1) with initial conition U(x, 0)=sin(x) an perioic bounary conitions. The exact solution is given by U(x, t)=sin(x+t). The numerical errors an orer of accuracy before an after post-processing can be foun in Table I. We clearly observe an accuracy of Dx k+1 before the post-processing an Dx k+1 or higher (closer to Dx k+ ) after it, when piecewise P k elements are use. We o not present the result with k=0 because for this case the results before the post-processing is alreay shown in [0] an the postprocessing oes not increase the orer of accuracy (k+1=k+1 for k=0). Example 5.. We compute the classical soliton solution of the nonlinear KV equation (1.3) in 1D: U t 3(U ) x +U xxx =0 (5.) Table I. U t +U xxx =0. U(x, 0)=sin(x). Perioic Bounary Conitions. L. Errors. Uniform Meshes with N Cells. LDG Methos with k=1,, 3. t=1. Before (B) an After (A) Post-Processing N=10 N=0 N=40 N=80 k error error orer error orer error orer 1 B 5.065E E E E A E E E E B.9084E E E E A.0588E E E E N=10 N=0 N=40 N=50 3 B 9.47E E E E A.816E E E E

18 44 Yan an Shu Table II. U t 3(U ) x +U xxx =0. U(x, 0)= sech (x). L. Errors. Uniform Meshes with N Cells. LDG Methos with k=1,, 3. t=0.5. Before (B) an After (A) Post-Processing N=80 N=160 N=30 N=640 k error error orer error orer error orer 1 B E E E E A 4.883E E E E B E E E E A 3.061E E E E N=80 N=160 N=30 N=500 3 B E E E E A 1.61E E E E with an initial conition u(x, 0)= sech (x) for 14[ x [ 16, see [0] for etails. The exact solution is given by U(x, t)= sech (x 4t). The numerical errors an orer of accuracy before an after post-processing can be foun in Table II. We clearly observe an accuracy of Dx k+1 before the post-processing an Dx k+1 or higher after it, when piecewise P k elements are use. This inicates that the post-processing technique is effective in enhancing orers of accuracy even for this nonlinear PDE. Example 5.3. We solve the linear bi-harmonic equation (1.4) in 1D: U t +U xxxx =0 (5.3) with initial conition U(x, 0)=sin(x) an perioic bounary conitions. The exact solution is given by U(x, t)=e t sin(x). The numerical errors an orer of accuracy before an after post-processing can be foun in Table 3. We clearly observe an accuracy of Dx k+1 before the post-processing an Dx k+1 or higher (closer to Dx k+ ) after it, when piecewise P k elements are use. We have use multiple precision FORTRAN to compute part of this problem, with an effective precision higher than the stanar ouble precision, to overcome some of the roun-off problems for this problem. If such multiple precision proceure is not use, then the post-processe error can only go own to aroun level an cannot ecrease further. The result in Table III verifies both the stability an accuracy of the local iscontinuous Galerkin metho evelope in Section 3 an also the effectiveness of the post-processing technique for such methos applie to this type of equations. It also strongly suggests that there is a better negative norm error estimate on the orer of Dx k+ for piecewise P k elements to this problem.

19 Local Discontinuous Galerkin Methos for Partial Differential Equations 45 Table III. U t +U xxxx =0. U(x, 0)=sin(x). Perioic Bounary Conitions. L. Errors. Uniform Meshes with N Cells. LDG Methos with k=0, 1,, 3. t=1. Before (B) an After (A) Post-Processing N=10 N=0 N=40 N=80 k error error orer error orer error orer 0 B 1.115E E E E B.038E-0 5.6E E E A E E E E B E E E E A E E E E B E E E E A 3.399E E E E Example 5.4. We solve the linear equation (1.5) in 1D: U t +U xxxxx =0 (5.4) with initial conition U(x, 0)=sin(x) an perioic bounary conitions. The exact solution is given by U(x, t)=sin(x t). The numerical errors an orer of accuracy before an after post-processing can be foun in Table IV. We clearly observe an accuracy of Dx k+1 before the postprocessing an Dx k+ (one orer higher than the expecte Dx k+1 ) after it, when piecewise P k elements are use. This strongly suggests that there is a better negative norm error estimate on the orer of Dx k+ (at least for k= an 3 among the teste cases) for piecewise P k elements to this problem. We have use multiple precision FORTRAN to compute part of this problem, with an effective precision higher than the stanar ouble precision, to overcome some of the roun-off problems for this problem. If such multiple precision proceure is not use, then the post-processe error can only go own to aroun 10 8 to 10 9 level an cannot ecrease further. The result in Table IV clearly verifies both the stability an accuracy of the local iscontinuous Galerkin metho evelope in Section 4 an also the effectiveness of the post-processing technique for such methos applie to this type of equations. 6. CONCLUDING REMARKS We have reviewe local iscontinuous Galerkin methos for solving PDEs with first, secon an thir spatial erivatives an evelope new local iscontinuous Galerkin methos for solving PDEs with fourth erivatives (the time epenent bi-harmonic equations) an those with fifth erivatives. We have esigne the numerical fluxes an prove that such

20 46 Yan an Shu Table IV. U t +U xxxxx =0. U(x, 0)=sin(x). Perioic Bounary Conitions. L. Errors. Uniform Meshes with N Cells. LDG Methos with k=0, 1,, 3. t=1. Before (B) an After (A) Post-Processing N=10 N=0 N=40 N=80 k error error orer error orer error orer 0 B 5.684E E E E B E E E E A 6.658E E E E B.9071E E E E A.384E E E E B 1.453E E E E A E E E E methos are L stable an satisfy cell entropy inequalities for general nonlinear PDEs. Although the iscussion is concentrate in 1D because of space limitation, the results can be easily generalize to multiple spatial imensions. Error estimates for linear equations similar to those offere in [11] an [0] can be obtaine along similar lines but are not given in this paper. PDEs with even higher spatial erivatives can also be hanle along similar lines. We also apply a post processing technique an emonstrate that the accuracy can be improve from Dx k+1 to Dx k+1 or higher when piecewise polynomials of egree k are use. In future work we will investigate efficient time marching of these local iscontinuous Galerkin methos without sacrificing its local property an parallel flexibility, an perform more numerical experiments with physically interesting problems. ACKNOWLEDGMENTS This research was supporte by ARO Grant DAAD , NSF Grants DMS an ECS , NASA Langley Grant NCC an Contract NAS while the secon author was in resience at ICASE, NASA Langley Research Center, Hampton, VA , an AFOSR Grant F REFERENCES 1. Bassi, F., an Rebay, S. (1997). A high-orer accurate iscontinuous finite element metho for the numerical solution of the compressible Navier Stokes equations. J. Comput. Phys. 131, Biswas, R., Devine, K. D., an Flaherty, J. (1994). Parallel, aaptive finite element methos for conservation laws. Appl. Numer. Math. 14, Cockburn, B. (1999). Discontinuous Galerkin methos for convection-ominate problems. In Barth, T. J., an Deconinck, H. (es.), High-Orer Methos for Computational Physics, Lecture Notes in Computational Science an Engineering, Vol. 9, Springer, pp

21 Local Discontinuous Galerkin Methos for Partial Differential Equations Cockburn, B., Hou, S., an Shu, C.-W. (1990). TVB Runge Kutta local projection iscontinuous Galerkin finite element metho for conservation laws IV: The multiimensional case. Math. Comp. 54, Cockburn, B., Karniaakis, G., an Shu, C.-W. (000). The evelopment of iscontinuous Galerkin methos. In Cockburn, B., Karniaakis, G., an Shu C.-W. (es.), Discontinuous Galerkin Methos: Theory, Computation an Applications, Lecture Notes in Computational Science an Engineering, Vol. 11, Springer, Part I: Overview, pp Cockburn, B., Lin, S.-Y., an Shu, C.-W. (1989). TVB Runge Kutta local projection iscontinuous Galerkin finite element metho for conservation laws III: One imensional systems. J. Comput. Phys. 84, Cockburn, B., Luskin, M., Shu, C.-W., an Süli, E. Enhance accuracy by post-processing for finite element methos for hyperbolic equations. Math. Comp. To appear. 8. Cockburn, B., an Shu, C.-W. (1991). The Runge Kutta local projection P 1 -iscontinuous- Galerkin finite element metho for scalar conservation laws. Math. Moel. Numer. Anal. (M AN) 5, Cockburn, B., an Shu, C.-W. (1989). TVB Runge Kutta local projection iscontinuous Galerkin finite element metho for scalar conservation laws II: General framework. Math. Comp. 5, Cockburn, B., an Shu, C.-W. (1998). TVB Runge Kutta local projection iscontinuous Galerkin finite element metho for scalar conservation laws V: Multiimensional systems. J. Comput. Phys. 141, Cockburn, B., an Shu, C.-W. (1998). The local iscontinuous Galerkin metho for time-epenent convection iffusion systems. SIAM J. Numer. Anal. 35, Cockburn, B., an Shu, C.-W. (001). Runge Kutta Discontinuous Galerkin methos for convection-ominate problems. J. Sci. Comput. 16, Dekker, K., an Verwer, J. G. (1984). Stability of Runge Kutta Methos for Stiff Nonlinear Differential Equations, North-Hollan. 14. Jiang, G.-S., an Shu, C.-W. (1994). On cell entropy inequality for iscontinuous Galerkin methos. Math. Comp. 6, Johnson, C., an Pitkäranta, J. (1986). An analysis of the iscontinuous Galerkin metho for a scalar hyperbolic equation, Math. Comp. 46, LeVeque, R. J. (1990). Numerical Methos for Conservation Laws, Birkhauser Verlag, Basel. 17. Shu, C.-W. (1987). TVB uniformly high-orer schemes for conservation laws. Math. Comp. 49, Shu, C.-W. (001). Different formulations of the iscontinuous Galerkin metho for the viscous terms. In Shi, Z.-C., Mu, M., Xue W., an Zou, J. (es.), Avances in Scientific Computing, Science Press, pp Shu, C.-W., an Osher, S. (1988). Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, Yan, J., an Shu, C.-W., A local iscontinuous Galerkin metho for KV type equations. SIAM J. Numer. Anal. to appear.