Discontinuous Galerkin Methods: General Approach and Stability

Transcription

1 Discontinuous Galerkin Metods: General Approac and Stability Ci-Wang Su Division of Applied Matematics, Brown University Providence, RI 9, USA Abstract In tese lectures, we will give a general introduction to te discontinuous Galerkin (DG) metods for solving time dependent, convection dominated partial differential equations (PDEs), including te yperbolic conservation laws, convection diffusion equations, and PDEs containing iger order spatial derivatives suc as te KdV equations and oter nonlinear dispersive wave equations. We will discuss cell entropy inequalities, nonlinear stability, and error estimates. Te important ingredient of te design of DG scemes, namely te adequate coice of numerical fluxes, will be explained in detail. Issues related to te implementation of te DG metod will also be addressed. Introduction Discontinuous Galerkin (DG) metods are a class of finite element metods using completely discontinuous basis functions, wic are usually cosen as piecewise polynomials. Since te basis functions can be completely discontinuous, tese metods ave te flexibility wic is not sared by typical finite element metods, suc as te allowance of arbitrary triangulation wit anging nodes, complete freedom in canging te polynomial degrees in eac element independent of tat in te neigbors (p adaptivity), and extremely local data structure (elements only communicate wit immediate neigbors regardless of te order of accuracy of te sceme) and te resulting embarrassingly ig parallel efficiency (usually more tan 99% for a fixed mes, and more tan 8% for a dynamic load balancing wit adaptive meses wic cange often during time evolution), see, e.g. [5]. A very good example to illustrate te capability of te discontinuous Galerkin

2 metod in -p adaptivity, efficiency in parallel dynamic load balancing, and excellent resolution properties is te successful simulation of te Rayleig-Taylor flow instabilities in [38]. Te first discontinuous Galerkin metod was introduced in 973 by Reed and Hill [37], in te framework of neutron transport, i.e. a time independent linear yperbolic equation. A major development of te DG metod is carried out by Cockburn et al. in a series of papers [4, 3,,, 5], in wic tey ave establised a framework to easily solve nonlinear time dependent problems, suc as te Euler equations of gas dynamics, using explicit, nonlinearly stable ig order Runge-Kutta time discretizations [44] and DG discretization in space wit exact or approximate Riemann solvers as interface fluxes and total variation bounded (TVB) nonlinear limiters [4] to acieve non-oscillatory properties for strong socks. Te DG metod as found rapid applications in suc diverse areas as aeroacoustics, electro-magnetism, gas dynamics, granular flows, magneto-ydrodynamics, meteorology, modeling of sallow water, oceanograpy, oil recovery simulation, semiconductor device simulation, transport of contaminant in porous media, turbomacinery, turbulent flows, viscoelastic flows and weater forecasting, among many oters. For more details, we refer to te survey paper [], and oter papers in tat Springer volume, wic contains te conference proceedings of te First International Symposium on Discontinuous Galerkin Metods eld at Newport, Rode Island in 999. Te lecture notes [8] is a good reference for many details, as well as te extensive review paper [7]. More recently, tere are two special issues devoted to te discontinuous Galerkin metod [8, 9], wic contain many interesting papers in te development of te metod in all aspects including algoritm design, analysis, implementation and applications. Time discretization In tese lectures, we will concentrate on te metod of lines DG metods, tat is, we do not discretize te time variable. Terefore, we will briefly discuss te issue of time discretization at te beginning. For yperbolic problems or convection dominated problems suc as ig Reynolds number Navier-Stokes equations, we often use a class of ig order nonlinearly stable Runge-Kutta time discretizations. A distinctive feature of tis class of time discretizations is tat tey are convex combinations of first order forward Euler steps, ence tey maintain strong stability properties in any semi-norm (total variation semi-norm, maximum norm, entropy condition, etc.) of te forward Euler step. Tus one only needs to prove nonlinear stability for te first order forward Euler step, wic is relatively easy in many situations (e.g. TVD scemes, see for example Section 3.. below), and one

3 automatically obtains te same strong stability property for te iger order time discretizations in tis class. Tese metods were first developed in [44] and [4], and later generalized in [] and []. Te most popular sceme in tis class is te following tird order Runge-Kutta metod for solving u t = L(u, t) were L(u, t) is a spatial discretization operator (it does not need to be, and often is not, linear!): u () = u n +ΔtL(u n,t n ) u () = 3 4 un + 4 u() + 4 ΔtL(u(),t n +Δt) (.) u n+ = 3 un + 3 u() + 3 ΔtL(u(),t n + Δt). Scemes in tis class wic are iger order or are of low storage also exist. For details, see te survey paper [43] and te review paper []. If te PDEs contain ig order spatial derivatives wit coefficients not very small, ten explicit time marcing metods suc as te Runge-Kutta metods described above suffer from severe time step restrictions. It is an important and active researc subject to study efficient time discretization for suc situations, wile still maintaining te advantages of te DG metods, suc as teir local nature and parallel efficiency. See, e.g. [46] for a study of several time discretization tecniques for suc situations. We will not furter discuss tis important issue toug in tese lectures. 3 Discontinuous Galerkin metod for conservation laws Te discontinuous Galerkin metod was first designed as an effective numerical metods for solving yperbolic conservation laws, wic may ave discontinuous solutions. In tis section we will discuss te algoritm formulation, stability analysis, and error estimates for te discontinuous Galerkin metod solving yperbolic conservation laws. 3. Two dimensional steady state linear equations We now present te details of te original DG metod in [37] for te two dimensional steady state linear convection equation au x + bu y = f(x, y), x, y (3.) 3

4 were a and b are constants. Witout loss of generality we assume a>, b>. Te equation (3.) is well-posed wen equipped wit te inflow boundary condition u(x, ) = g (x), x and u(,y)=g (y), y. (3.) For simplicity, we assume a rectangular mes to cover te computational domain [, ], consisting of cells,j = {(x, y) : x i for i N x and j N y,were =x <x3 x x i+, y j < <x Nx+ y y j+ } = and =y <y3 < <y Ny+ = are discretizations in x and y over [, ]. We also denote and Δx i = x i+ x i, i N x ; Δy j = y j+ ( =max max i N x Δx i, ) max Δy j. j N y y j, j N y ; We assume te mes is regular, namely tere is a constant c> independent of suc tat Δx i c, i N x ; Δy j c, j N y. We define a finite element space consisting of piecewise polynomials V k = { } v : v Ii,j P k (,j ); i N x, j N y (3.3) were P k (,j ) denotes te set of polynomials of degree up to k defined on te cell,j. Notice tat functions in V k may be discontinuous across cell interfaces. Te discontinuous Galerkin (DG) metod for solving (3.) is defined as follows: find te unique function u V k suc tat, for all test functions v V k and all i N x and j N y,weave,j (au (v ) x + bu (v ) y ) dxdy + a y j+ û y j (x i+,y)v (x,y)dy i+ a y j+ û y j (x i,y)v (x +,y)dy + b x i+ i û x i (x, y j+ )v (x, y )dx (3.4) j+ b x i+ û x i (x, y j )v (x, y + )dx =. j 4

5 Here, û is te so-called numerical flux, wic is a single valued function defined at te cell interfaces and in general depends on te values of te numerical solution u from bot sides of te interface, since u is discontinuous tere. For te simple linear convection PDE (3.), te numerical flux can be cosen according to te upwind principle, namely û (x i+,y)=u (x,y), û i+ (x, y j+ )=u (x, y ). j+ Notice tat, for te boundary cell i =, te numerical flux for te left edge is defined using te given boundary condition û (x,y)=g (y). Likewise, for te boundary cell j =, te numerical flux for te bottom edge is is defined by û (x, y )=g (x). We now look at te implementation of te sceme (3.4). If a local basis of P k (,j )is cosen and denoted as ϕ l i,j (x, y) forl =,,,K =(k +)(k +)/, we can express te numerical solution as u (x, y) = K l= and we sould solve for te coefficients u l i,j ϕl i,j (x, y), (x, u i,j = u i,j. u K i,j y),j wic, according to te sceme (3.4), satisfies te linear equation A i,j u i,j = rs (3.5) were A i,j is a K K matrix wose (l, m)-t entry is given by ( ) a l,m i,j = aϕ m i,j (x, y)(ϕ l i,j(x, y)) x + bϕ m i,j(x, y)(ϕ l i,j(x, y)) y dxdy (3.6),j +a yj+ y j ϕ m i,j(x i+,y)ϕ l i,j(x i+,y)dy + b xi+ and te l-t entry of te rigt-and-side vector is given by rs l = a yj+ y j u (x i,y)ϕ l i,j (x i,y)dy + b 5 ϕ m i,j(x, y j+ )ϕ l i,j(x, y j+ )dx, x i xi+ u (x, y )ϕ l j i,j (x, y j )dx x i

6 wic depends on te information of u in te left cell,j and te bottom cell,j, if tey are in te computational domain, or on te boundary condition, if one or bot of tese cells are outside te computational domain. It is easy to verify tat te matrix A i,j in (3.5) wit entries given by (3.6) is invertible, ence te numerical solution u in te cell,j can be easily obtained by solving te small linear system (3.5), once te solution at te left and bottom cells,j and,j are already known, or if one or bot of tese cells are outside te computational domain. Terefore, we can obtain te numerical solution u in te following ordering: first we obtain it in te cell I,, since bot its left and bottom boundaries are equipped wit te prescribed boundary conditions (3.). We ten obtain te solution in te cells I, and I,.ForI,, te numerical solution u in its left cell I, is already available, and its bottom boundary is equipped wit te prescribed boundary condition (3.). Similar argument goes for te cell I,. Te next group of cells to be solved are I 3,, I,, I,3. It is clear tat we can obtain te solution u sequentially in tis way for all cells in te computational domain. Clearly, tis metod does not involve any large system solvers and is very easy to implement. In [5], Lesaint and Raviart proved tat tis metod is convergent wit te optimal order of accuracy, namely O( k+ ), in L norm, wen piecewise tensor product polynomials of degree k are used as basis functions. Numerical experiments indicate tat te convergence rate is also optimal wen te usual piecewise polynomials of degree k (3.3) are used instead. Notice tat, even toug te metod (3.4) is designed for te steady state problem (3.), it can be easily used on initial-boundary value problems of linear time dependent yperbolic equations: we just need to identify te time variable t as one of te spatial variables. It is also easily generalizable to iger dimensions. Te metod described above can be easily designed and efficiently implemented on arbitrary triangulations. L error estimates of O( k+/ )werek is again te polynomial degree and is te mes size can be obtained wen te solution is sufficiently smoot, for arbitrary meses, see, e.g. [4]. Tis estimate is actually sarp for te most general situation [33], owever in many cases te optimal O( k+ ) error bound can be proved [39, 9]. In actual numerical computations, one almost always observe te optimal O( k+ ) accuracy. Unfortunately, even toug te metod (3.4) is easy to implement, accurate, and efficient, it cannot be easily generalized to linear systems, were te caracteristic information comes from different directions, or to nonlinear problems, were te caracteristic wind direction depends on te solution itself. 6

7 3. One dimensional time dependent conservation laws Te difficulties mentioned at te end of te last subsection can be by-passed wen te DG discretization is only used for te spatial variables, and te time discretization is acieved by te explicit Runge-Kutta metods suc as (.). Tis is te approac of te so-called Runge-Kutta discontinuous Galerkin (RKDG) metod [4, 3,,, 5]. We start our discussion wit te one dimensional conservation law u t + f(u) x =. (3.7) As before, we assume te following mes to cover te computational domain [, ], consisting of cells =[x i,x i+ ], for i N, were We again denote Δx i = x i+ =x <x3 < <x N+ =. x i, i N; = max Δx i. i N We assume te mes is regular, namely tere is a constant c> independent of suc tat Δx i c, i N. We define a finite element space consisting of piecewise polynomials V k = { v : v Ii P k ( ); i N } (3.8) were P k ( ) denotes te set of polynomials of degree up to k defined on te cell.te semi-discrete DG metod for solving (3.7) is defined as follows: find te unique function u = u (t) V k suc tat, for all test functions v V k and all i N, weave (u ) t (v )dx f(u )(v ) x dx + ˆf i+ v (x ) ˆf i+ i v (x + )=. I i i (3.9) Here, ˆf i+ is again te numerical flux, wic is a single valued function defined at te cell interfaces and in general depends on te values of te numerical solution u from bot sides of te interface ˆf i+ = ˆf(u (x,t),u i+ (x +,t)). i+ We use te so-called monotone fluxes from finite difference and finite volume scemes for solving conservation laws, wic satisfy te following conditions: Consistency: ˆf(u, u) =f(u); 7

8 Continuity: ˆf(u,u + ) is at least Lipscitz continuous wit respect to bot arguments u and u +. Monotonicity: ˆf(u,u + ) is a non-decreasing function of its first argument u and a non-increasing function of its second argument u +. Symbolically ˆf(, ). Well known monotone fluxes include te Lax-Friedrics flux te Godunov flux ˆf LF (u,u + )= ˆf God (u,u + )= and te Engquist-Oser flux ˆf EO = u ( f(u )+f(u + ) α(u + u ) ), α =max f (u) ; u { minu u u + f(u), if u <u + max u + u u f(u), if u u + ; max(f (u), )du + u + We refer to, e.g., [6] for more details about monotone fluxes. 3.. Cell entropy inequality and L stability min(f (u), )du + f(). It is well known tat weak solutions of (3.7) may not be unique and te unique, pysically relevant weak solution (te so-called entropy solution) satisfies te following entropy inequality U(u) t + F (u) x (3.) in distribution sense, for any convex entropy U(u) satisfying U (u) and te corresponding entropy flux F (u) = u U (u)f (u)du. It will be nice if a numerical approximation to (3.7) also sares a similar entropy inequality as (3.). It is usually quite difficult to prove a discrete entropy inequality for finite difference or finite volume scemes, especially for ig order scemes and wen te flux function f(u) in (3.7) is not convex or concave, see, e.g. [8, 3]. However, it turns out tat it is easy to prove tat te DG sceme (3.9) satisfies a cell entropy inequality [3]. Proposition 3.. Te solution u to te semi-discrete DG sceme (3.9) satisfies te following cell entropy inequality d U(u ) dx + dt ˆF i+ ˆF i (3.) 8

9 for te square entropy U(u) = u, for some consistent entropy flux ˆF i+ = ˆF (u (x,t),u i+ (x +,t)) i+ satisfying ˆF (u, u) =F (u). Proof: We introduce a sort-and notation B i (u ; v )= (u ) t (v )dx If we take v = u in te sceme (3.9), we obtain B i (u ; u )= (u ) t (u )dx f(u )(v ) x dx + ˆf i+ v (x i+ f(u )(u ) x dx+ ˆf i+ u (x i+ ) ˆf i v (x + ). (3.) i ) ˆf i u (x + )=. (3.3) i If we denote F (u) = u f(u)du, ten (3.3) becomes B i (u ; u )= U(u ) t dx F (u (x ))+ F (u i+ (x + ))+ ˆf i i+ u (x ) ˆf I i+ i u (x + )= i i or B i (u ; u )= U(u ) t dx + ˆF i+ ˆF i +Θ i = (3.4) were and Θ i ˆF i+ = F (u (x )) + ˆf i+ i+ u (x ) (3.5) i+ = F (u (x )) + ˆf i i u (x )+ F (u i (x + )) ˆf i i u (x + ). (3.6) i It is easy to verify tat te numerical entropy flux ˆF defined by (3.5) is consistent wit te entropy flux F (u) = u U (u)f (u)du for U(u) = u.itisalsoeasytoverify Θ= F (u )+ ˆfu + F (u + ) ˆfu + =(u+ u )( F (ξ) ˆf) were we ave dropped te subscript i since all quantities are evaluated tere in. A mean value teorem is applied and ξ is a value between u and u +, and we ave Θ i used te fact F (ξ) =f(ξ) and te monotonicity of te flux function ˆf to obtain te last inequality. Tis finises te proof of te cell entropy inequality (3.). We note tat te proof does not depend on te accuracy of te sceme, namely it olds for te piecewise polynomial space (3.8) wit any degree k. Also, te same proof can be given for te multi-dimensional DG sceme on any triangulation. 9

10 Te cell entropy inequality trivially implies an L stability of te numerical solution. Proposition 3.. For periodic or compactly supported boundary conditions, te solution u to te semi-discrete DG sceme (3.9) satisfies te following L stability or d dt (u ) dx (3.7) u (,t) u (, ). (3.8) Here and below, an unmarked norm is te usual L norm. Proof: We simply sum up te cell entropy inequality (3.) over i. Te flux terms telescope and tere is no boundary term left because of te periodic or compact supported boundary condition. (3.7), and ence (3.8), is now immediate. Notice tat bot te cell entropy inequality (3.) and te L stability (3.7) are valid even wen te exact solution of te conservation law (3.7) is discontinuous. 3.. Limiters and total variation stability For discontinuous solutions, te cell entropy inequality (3.) and te L stability (3.7), altoug elpful, are not enoug to control spurious numerical oscillations near discontinuities. In practice, especially for problems containing strong discontinuities, we often need to apply nonlinear limiters to control tese oscillations and to obtain provable total variation stability. For simplicity, we first consider te forward Euler time discretization of te semidiscrete DG sceme (3.9). Starting from a preliminary solution u n,pre n (for te initial condition, u,pre condition u(, ) into V k function u n V k for all test functions v V k u n+,pre u n Δt V k at time level is taken to be te L projection of te analytical initial ), we would like to limit or pre-process it to obtain a new before advancing it to te next time level: find un+,pre V k suc tat, and all i N, weave v dx f(u n )(v ) x dx + ˆf n v i+ (x ) ˆf n v i+ i (x + ) = (3.9) I i i were Δt = t n+ t n is te time step. Tis limiting procedure to go from u n,pre sould satisfy te following two conditions: to u n It sould not cange te cell averages of u n,pre. Tat is, te cell averages of u n and are te same. Tis is for te conservation property of te DG metod. u n,pre

11 It sould not affect te accuracy of te sceme in smoot regions. Tat is, in te smoot regions tis limiter does not cange te solution, u n (x). (x) =un,pre Tere are many limiters discussed in te literature, and tis is still an active researc area, especially for multi-dimensional systems, see, e.g. [6]. We will only present an example [3] ere. We denote te cell average of te solution u as and furter denote ū i = Δx i u dx (3.) ũ i = u (x ) ū i+ i, ũ i =ū i u (x + ). (3.) i Te limiter sould not cange ū i but it may cange ũ i and/or ũ i. minmod limiter [3] canges ũ i and ũ i into In particular, te ũ (mod) i = m(ũ i, Δ + ū i, Δ ū i ), ũ (mod) i = m(ũ i, Δ + ū i, Δ ū i ), (3.) were Δ + ū i =ū i+ ū i, Δ ū i =ū i ū i, and te minmod function m is defined by { s min( a,, a m(a,,a l )= l ), if s = sign(a )= sign(a l );, oterwise. (3.3) Te limited function u (mod) is ten recovered to maintain te old cell average (3.) and te new point values given by (3.), tat is u (mod) (x )=ū i+ i +ũ (mod) i, u (mod) (x + )=ū i i ũ (mod) i (3.4) by te definition (3.). Tis recovery is unique for P k polynomials wit k. For k>, we ave extra freedom in obtaining u (mod). We could for example coose u (mod) to be te unique P polynomial satisfying (3.) and (3.4). Before discussing te total variation stability of te DG sceme (3.9) wit te preprocessing, we first present a simple Lemma due to Harten []. Lemma 3. (Harten) If a sceme can be written in te form u n+ i = u n i + C i+ Δ + u n i D i Δ u n i (3.5)

12 wit periodic or compactly supported boundary conditions, were C i+ and D i may be nonlinear functions of te grid values u n j for j = i p,,i+ q wit some p, q, satisfying C i+, D i+, C i+ + D i+ ten te sceme is TVD TV(u n+ ) TV(u n ) were te total variation seminorm is defined by, i (3.6) TV(u) = i Δ + u i. Proof: Taking te forward difference operation on (3.5) yields Δ + u n+ i = Δ + u n i + C i+ 3 = ( C i+ Δ + u n i+ C i+ D i+ )Δ + u n i + C i+ 3 Δ + u n i D i+ Δ + u n i + D i Δ u n i Δ u n i. Δ + u n i+ + D i Tanks to (3.6) and using te periodic or compactly supported boundary condition, we can take te absolute value on bot sides of te above equality and sum up over i to obtain i Δ + u n+ i i Tis finises te proof. ( C i+ D i+ ) Δ + u n i + i C i+ Δ + u n i + i We define te total variation in te means semi-norm, or TVM, as D i+ Δ + u n i = i Δ + u n i TVM(u )= i Δ + ū i. We ten ave te following stability result. Proposition 3.3. For periodic or compactly supported boundary conditions, te solution u n of te DG sceme (3.9), wit te pre-processing by te limiter, is total variation diminising in te means (TVDM), tat is TVM(u n+ ) TVM(u n ). (3.7) Proof: Taking v =forx in (3.9) and dividing bot sides by Δx i, we obtain, by noticing (3.4), ( ū n+,pre i =ū i λ i ˆf(ūi +ũ i, ū i+ ũ i+ ) ˆf(ū ) i +ũ i, ū i ũ i )

13 were λ i = Δt Δx i, and all quantities on te rigt and side are at te time level n. Wecan write te rigt and side of te above equality in te Harten form (3.5) if we define C i+ as follows and D i C i+ D i = λ i ˆf(ūi +ũ i, ū i+ ũ i+ ) ˆf(ū i +ũ i, ū i ũ i ) Δ + ū i, (3.8) = λ i ˆf(ūi +ũ i, ū i ũ i ) ˆf(ū i +ũ i, ū i ũ i ) Δ ū i. We now need to verify tat C i+ can write C i+ as in wic C i+ and D i = λ i ˆf ( ũ i+ Δ + ū i + defined in (3.8) satisfy (3.6). Indeed, we ũ i Δ + ū i ) (3.9) λ i ˆf = λ i ˆf(ūi +ũ i, ū i+ ũ i+ ) ˆf(ū i +ũ i, ū i ũ i ) (ū i+ ũ i+ ) (ū i ũ i ) λ i L (3.3) were we ave used te monotonicity and Lipscitz continuity of ˆf, andl is te Lipscitz constant of ˆf wit respect to its second argument. Also, since u n is te pre-processed solution by te minmod limiter, ũ i+ and ũ i are te modified values defined by (3.), ence ũ i+, ũ i. (3.3) Δ + ū i Δ + ū i Terefore, we ave, by (3.9), (3.3) and (3.3), Similarly, we can sow tat C i+ D i+ λ i L. λ i+ L were L is te Lipscitz constant of ˆf wit respect to its first argument. Tis proves (3.6) if we take te time step so tat λ (L + L ) were λ =max i λ i. Te TVDM property (3.7) ten follows from te Harten Lemma and te fact tat te limiter does not cange cell averages, ence TVM(u n+ )=TVM(u n+,pre ). 3

14 Even toug te previous proposition is proved only for te first order Euler forward time discretization, te special TVD (or strong stability preserving, SSP) Runge-Kutta time discretizations [44, ] allow us to obtain te same stability result for te fully discretized RKDG scemes. Proposition 3.4. Under te same conditions as tose in Proposition 3.3, te solution u n of te DG sceme (3.9), wit te Euler forward time discretization replaced by any SSP Runge-Kutta time discretization [] suc as (.), is TVDM. We still need to verify tat te limiter (3.) does not affect accuracy in smoot regions. If u is an approximation to a (locally) smoot function u, tenasimpletaylor expansion gives wile ũ i = u x(x i )Δx i + O( ), ũ i = u x(x i )Δx i + O( ), Δ + ū i = u x(x i )(Δx i +Δx i+ )+O( ), Δ ū i = u x(x i )(Δx i +Δx i )+O( ). Clearly, wen we are in a smoot and monotone region, namely wen u x (x i )isaway from zero, te first argument in te minmod function (3.) is of te same sign as te second and tird arguments and is smaller in magnitude (for a uniform mes it is about alf of teir magnitude), wen is small. Terefore, since te minmod function (3.3) picks te smallest argument (in magnitude) wen all te arguments are of te same sign, te modified values ũ (mod) i and ũ (mod) i in (3.) will take te unmodified values ũ i and ũ i, respectively. Tat is, te limiter does not affect accuracy in smoot, monotone regions. On te oter and, te TVD limiter (3.) does kill accuracy at smoot extrema. Tis is demonstrated by numerical results and is a consequence of te general results about TVD scemes, tat tey are at most second order accurate for smoot but nonmonotone solutions [3]. Terefore, in practice we often use a total variation bounded (TVB) corrected limiter { a, if a m(a,,a l )= M ; m(a,,a l ), oterwise instead of te original minmod function (3.3), were te TVB parameter M as to be cosen adequately [3]. Te DG sceme would ten be total variation bounded in te means (TVBM) and uniformly ig order accurate for smoot solutions. We will not discuss more details ere and refer te readers to [3]. We would like to remark tat te limiters discussed in tis subsection are first used for finite volume scemes [3]. Wen discussing limiters, te DG metods and finite volumes scemes ave many similarities. 4

15 3..3 Error estimates for smoot solutions If we assume te exact solution of (3.7) is smoot, we can obtain optimal L error estimates. Suc error estimates can be obtained for te general nonlinear conservation law (3.7) and for fully discretized RKDG metods, see [58]. However, for simplicity we will give ere te proof only for te semi-discrete DG sceme and te linear version of (3.7): u t + u x = (3.3) for wic te monotone flux is taken as te simple upwind flux ˆf(u,u + )=u.ofcourse te proof is te same for u t + au x = wit any constant a. Proposition 3.5. Te solution u of te DG sceme (3.9) for te PDE (3.3) wit a smoot solution u satisfies te following error estimate u u C k+ (3.33) were C depends on u and its derivatives but is independent of. Proof: Te DG sceme (3.9), wen using te notation in (3.), can be written as B i (u ; v ) = (3.34) for all v V and for all i. It is easy to verify tat te exact solution of te PDE (3.3) also satisfies B i (u; v ) = (3.35) for all v V and for all i. Subtracting (3.34) from (3.35) and using te linearity of B i wit respect to its first argument, we obtain te error equation B i (u u ; v ) = (3.36) for all v V and for all i. We now define a special projection P into V. For a given smoot function w, te projection Pw is te unique function in V wic satisfies, for eac i, (Pw(x) w(x))v (x)dx = v P k ( ); Pw(x )=w(x i+ i+ ). (3.37) I i Standard approximation teory [7] implies, for a smoot function w, Pw(x) w(x) C k+ (3.38) were ere and below C is a generic constant depending on w and its derivatives but independent of (wic may not ave te same value in different places). In particular, 5

16 in (3.38), C = C w H k+ were w H k+ is te standard Sobolev (k +)normand C is a constant independent of w. We now take: v = Pu u (3.39) in te error equation (3.36), and denote e = Pu u, ε = u Pu (3.4) to obtain B i (e ; e )= B i (ε ; e ). (3.4) For te left and side of (3.4), we use te cell entropy inequality (see (3.4)) to obtain B i (e ; e )= d (e ) dx + dt ˆF i+ ˆF i +Θ i (3.4) were Θ i. As to te rigt and side of (3.4), we first write out all te terms B i (ε ; e )= (ε ) t e dx + ε (e ) x dx (ε ) (e i+ ) +(ε i+ ) (e i ) +. i+ I i Noticing te properties (3.37) of te projection P, weave ε (e ) x dx = because (e ) x is a polynomial of degree at most k, and (ε ) = u i+ i+ (Pu) = i+ for all i. Terefore, te rigt and side of (3.4) becomes B i (ε ; e )= (ε ) t e dx ( ) ((ε ) t ) dx + (e ) dx (3.43) Plugging (3.4) and (3.43) into te equality (3.4), summing up over i, and using te approximation result (3.38), we obtain d (e ) dx (e ) dx + C k+. dt A Gronwall s inequality, te fact tat te initial error u(, ) u (, ) C k+ (usually te initial condition u (, ) is taken as te L projection of te analytical initial condition u(, )), and te approximation result (3.38) finally give us te error estimate (3.33). 6

17 3.3 Comments for multi-dimensional cases Even toug we ave only discussed te two dimensional steady state and one dimensional time dependent cases in previous subsections, Most of te results also old for multi-dimensional cases wit arbitrary triangulations. For example, te semi-discrete DG metod for te two dimensional time dependent conservation law u t + f(u) x + g(u) y = (3.44) is defined as follows. Te computational domain is partitioned into a collection of cells i, wic in D could be rectangles, triangles, etc., and te numerical solution is a polynomial of degree k in eac cell i. Te degree k could cange wit te cell, and tere is no continuity requirement of te two polynomials along an interface of two cells. Tus, instead of only one degree of freedom per cell as in a finite volume sceme, namely te cell average of te solution, tere are now K = (k+)(k+) degrees of freedom per cell for a DG metod using piecewise k-t degree polynomials in D. Tese K degrees of freedom are cosen as te coefficients of te polynomial wen expanded in a local basis. One could use a locally ortogonal basis to simplify te computation, but tis is not essential. Te DG metod is obtained by multiplying (3.44) by a test function v(x, y) (wicis also a polynomial of degree k in te cell), integrating over te cell j, and integrating by parts: d u(x, y, t)v(x, y)dxdy F (u) vdxdy+ F (u) nvds = (3.45) dt j j j were F =(f,g), and n is te outward unit normal of te cell boundary j. Te line integral in (3.45) is typically discretized by a Gaussian quadrature of sufficiently ig order of accuracy, q F nvds j ω k F (u(g k,t)) nv(g k ), j k= were F (u(g k,t)) n is replaced by a numerical flux (approximate or exact Riemann solvers). For scalar equations te numerical flux can be taken as any of te monotone fluxes discussed in Section 3. along te normal direction of te cell boundary. For example, one could use te simple Lax-Friedrics flux, wic is given by F (u(g k,t)) n [( F (u (G k,t)) + F (u + (G k,t)) ) n α ( u + (G k,t) u (G k,t) )] were α is taken as an upper bound for te eigenvalues of te Jacobian in te n direction, and u and u + are te values of u inside te cell j and outside te cell j (inside te 7

18 neigboring cell) at te Gaussian point G k. v(g k ) is taken as v (G k ), namely te value of v inside te cell j at te Gaussian point G k. Te volume integral term j F (u) vdxdy can be computed eiter by a numerical quadrature or by a quadrature free implementation [] for special systems suc as te compressible Euler equations. Notice tat if a locally ortogonal basis is cosen, te time derivative term d dt j u(x, y, t)v(x, y)dxdy would be explicit and tere is no mass matrix to invert. However, even if te local basis is not ortogonal, one still only needs to invert a small K K local mass matrix (by and) and tere is never a global mass matrix to invert as in a typical finite element metod. For scalar equations (3.44), te cell entropy inequality described in Proposition 3. olds for arbitrary triangulation. Te limiter described in Section 3.. can also be defined for arbitrary triangulation, see []. Instead of te TVDM property given in Proposition 3.3, for multi-dimensional cases we can prove te maximum norm stability of te limited sceme, see []. Te optimal error estimate given in Proposition 3.5 can be proved for tensor product meses and basis functions, and for certain specific triangulations wen te usual piecewise k-t degree polynomial approximation spaces are used [39, 9]. For te most general cases, an L error estimate of alf an order lower O( k+ )canbeproved [4], wic is actually sarp [33]. For nonlinear yperbolic equations including symmetrizable systems, if te solution of te PDE is smoot, L error estimates of O( k+/ +Δt )wereδt is te time step can be obtained for te fully discrete Runge-Kutta discontinuous Galerkin metod wit second order Runge-Kutta time discretization. For upwind fluxes te optimal O( k+ +Δt ) error estimate can be obtained. See [58, 59]. As an example of te excellent numerical performance of te RKDG sceme, we sow in Figures 3. and 3. te solution of te second order (piecewise linear) and sevent order (piecewise polynomial of degree 6) DG metods for te linear transport equation u t + u x =, or u t + u x + u y =, on te domain (, π) (,T)or(, π) (,T) wit te caracteristic function of te interval ( π, 3π )ortesquare(π, 3π ) as initial condition and periodic boundary conditions [7]. Notice tat te solution is for a very long time, t = π (5 time periods), wit a relatively coarse mes. We can see tat te second order sceme smears te fronts, owever te sevent order sceme maintains te sape of te solution almost as well as te initial condition! Te excellent performance can be acieved by te DG metod on multi-dimensional linear systems using unstructured meses, ence it is a very good metod for solving, e.g. Maxwell equations of electromagnetism and linearized Euler equations of aeroacoustics. To demonstrate tat te DG metod also works well for nonlinear systems, we sow in Figure 3.3 te DG solution of te forward facing step problem by solving te compressible 8

19 k=, t=π, solid line: exact solution; dased line / squares: numerical solution u x k=6, t=π, solid line: exact solution; dased line / squares: numerical solution u x Figure 3.: Transport equation: Comparison of te exact and te RKDG solutions at T = π wit second order (P, left) and sevent order (P 6, rigt) RKDG metods. One dimensional results wit 4 cells, exact solution (solid line) and numerical solution (dased line and symbols, one point per cell). P P u u y x y x Figure 3.: Transport equation: Comparison of te exact and te RKDG solutions at T = π wit second order (P, left) and sevent order (P 6, rigt) RKDG metods. Two dimensional results wit 4 4 cells. 9

20 . Rectangles P, Δ x = Δ y = / Rectangles P, Δ x = Δ y = / Figure 3.3: Forward facing step. Zoomed-in region. Δx =Δy = 3.Top:P elements; bottom: P elements. Euler equations of gas dynamics [5]. We can see tat te roll-ups of te contact line caused by a pysical instability are resolved well, especially by te tird order DG sceme. In summary, we can say te following about te discontinuous Galerkin metods for conservation laws:. Tey can be used for arbitrary triangulation, including tose wit anging nodes. Moreover, te degree of te polynomial, ence te order of accuracy, in eac cell can be independently decided. Tus te metod is ideally suited for -p (mes size and order of accuracy) refinements and adaptivity.. Te metods ave excellent parallel efficiency. Even wit space time adaptivity and load balancing te parallel efficiency can still be over 8%, see [38]. 3. Tey sould be te metods of coice if geometry is complicated or if adaptivity is important, especially for problems wit long time evolution of smoot solutions.

21 4. For problems containing strong socks, te nonlinear limiters are still less robust tan te advanced WENO pilosopy. Tere is a parameter (te TVB constant) for te user to tune for eac problem, see [3,, 5]. For rectangular meses te limiters work better tan for triangular ones. In recent years, WENO based limiters ave been investigated [35, 34, 36]. 4 Discontinuous Galerkin metod for convection diffusion equations In tis section we discuss te discontinuous Galerkin metod for time dependent convection diffusion equations d d d u t + f i (u) xi (a ij (u)u xj ) xi = (4.) i= i= were (a ij (u)) is a symmetric, semi-positive definite matrix. Tere are several different formulations of discontinuous Galerkin metods for solving suc equations, e.g. [, 4, 6, 9, 45], owever in tis section we will only discuss te local discontinuous Galerkin (LDG) metod [6]. For equations containing iger order spatial derivatives, suc as te convection diffusion equation (4.), discontinuous Galerkin metods cannot be directly applied. Tis is because te solution space, wic consists of piecewise polynomials discontinuous at te element interfaces, is not regular enoug to andle iger derivatives. Tis is a typical non-conforming case in finite elements. A naive and careless application of te discontinuous Galerkin metod directly to te eat equation containing second derivatives could yield a metod wic beaves nicely in te computation but is inconsistent wit te original equation and as O() errors to te exact solution [7, 57]. Te idea of local discontinuous Galerkin metods for time dependent partial differential equations wit iger derivatives, suc as te convection diffusion equation (4.), is to rewrite te equation into a first order system, ten apply te discontinuous Galerkin metod on te system. A key ingredient for te success of suc metods is te correct design of interface numerical fluxes. Tese fluxes must be designed to guarantee stability and local solvability of all te auxiliary variables introduced to approximate te derivatives of te solution. Te local solvability of all te auxiliary variables is wy te metod is called a local discontinuous Galerkin metod in [6]. Te first local discontinuous Galerkin metod was developed by Cockburn and Su [6], for te convection diffusion equation (4.) containing second derivatives. Teir work was motivated by te successful numerical experiments of Bassi and Rebay [3] for te compressible Navier-Stokes equations. j=

22 In te following we will discuss te stability and error estimates for te LDG metod for convection diffusion equations. We present details only for te one dimensional case and will mention briefly te generalization to multi-dimensions in Section LDG sceme formulation We consider te one dimensional convection diffusion equation u t + f(u) x =(a(u)u x ) x (4.) wit a(u). We rewrite tis equation as te following system u t + f(u) x =(b(u)q) x, q B(u) x = (4.3) were b(u) = a(u), B(u) = u b(u)du. (4.4) Te finite element space is still given by (3.8). Te semi-discrete LDG sceme is defined as follows. Find u,q V k suc tat, for all test functions v,p V k and all i N, we ave (u ) t (v )dx (f(u ) b(u )q )(v ) x dx +( ˆf ˆbˆq) i+ (v ) ( ˆf ˆbˆq) i+ i (v ) + =, (4.5) i q p dx + B(u )(p ) x dx ˆB i+ (p ) + ˆB i+ i (p ) + =. i Here, all te at terms are te numerical fluxes, namely single valued functions defined at te cell interfaces wic typically depend on te discontinuous numerical solution from bot sides of te interface. We already know from Section 3 tat te convection flux ˆf sould be cosen as a monotone flux. However, te upwinding principle is no longer a valid guiding principle for te design of te diffusion fluxes ˆb, ˆq and ˆB. In [6], sufficient conditions for te coices of tese diffusion fluxes to guarantee te stability of te sceme (4.5) are given. Here, we will discuss a particularly attractive coice, called alternating fluxes, defined as ˆb = B(u + ) B(u ) u + u, ˆq = q +, ˆB = B(u ). (4.6) Te important point is tat ˆq and ˆB sould be cosen from different directions. Tus, te coice B(u ˆb + = ) B(u ) u +, ˆq = q u, ˆB = B(u + )

23 is also fine. Notice tat, from te second equation in te sceme (4.5), we can solve q explicitly and locally (in cell )intermsofu, by inverting te small mass matrix inside te cell. Tis is te reason tat te metod is referred to as te local discontinuous Galerkin metod. 4. Stability analysis Similar to te case for yperbolic conservation laws, we ave te following cell entropy inequality for te LDG metod (4.5). Proposition 4.. Te solution u, q to te semi-discrete LDG sceme (4.5) satisfies te following cell entropy inequality d (u ) dx + (q ) dx + dt ˆF i+ ˆF i (4.7) for some consistent entropy flux ˆF i+ = ˆF (u (x,t),q i+ (x,t); u i+ (x +,t),q i+ (x + )) i+ satisfying ˆF (u, u) =F (u) ub(u)q were, as before, F (u) = u uf (u)du. Proof: We introduce a sort-and notation B i (u,q ; v,p ) = (u ) t (v )dx (f(u ) b(u )q )(v ) x dx +( ˆf ˆbˆq) i+ (v ) ( ˆf ˆbˆq) i+ i (v ) + (4.8) i + q p dx + B(u )(p ) x dx ˆB i+ (p ) + ˆB i+ i (p ) +. I i i If we take v = u, p = q in te sceme (4.5), we obtain B i (u,q ; u,q ) = (u ) t (u )dx (f(u ) b(u )q )(u ) x dx +( ˆf ˆbˆq) i+ (u ) ( ˆf ˆbˆq) i+ i (u ) + i + (q ) dx + B(u )(q ) x dx ˆB i+ (q ) i+ I i =. + ˆB i (q ) + i (4.9) 3

24 If we denote F (u) = u f(u)du, ten (4.9) becomes B i (u,q ; u,q )= d (u ) dx + dt (q ) dx + ˆF i+ ˆF i +Θ i = (4.) were ˆF = F (u )+ ˆfu ˆbq + u (4.) and Θ= F (u )+ ˆfu + F (u + ) ˆfu +, (4.) were we ave used te definition of te numerical fluxes (4.6). Notice tat we ave omitted te subindex i in te definitions of ˆF and Θ. It is easy to verify tat te numerical entropy flux ˆF defined by (4.) is consistent wit te entropy flux F (u) ub(u)q. As Θ in (4.) is te same as tat in (3.6) for te conservation law case, we readily ave Θ. Tis finises te proof of (4.7). We again note tat te proof does not depend on te accuracy of te sceme, namely it olds for te piecewise polynomial space (3.8) wit any degree k. Also, te same proof can be given for multi-dimensional LDG scemes on any triangulation. As before, te cell entropy inequality trivially implies an L stability of te numerical solution. Proposition 4.. For periodic or compactly supported boundary conditions, te solution u, q to te semi-discrete LDG sceme (4.5) satisfies te following L stability or d dt u (,t) + (u ) dx + t (q ) dx (4.3) q (,τ) dτ u (, ). (4.4) Notice tat bot te cell entropy inequality (4.7) and te L stability (4.3) are valid regardless of weter te convection diffusion equation (4.) is convection dominate or diffusion dominate and regardless of weter te exact solution of te PDE is smoot or not. Te diffusion coefficient a(u) can be degenerate (equal to zero) in any part of te domain. Te LDG metod is particularly attractive for convection dominated convection diffusion equations, wen traditional continuous finite element metods may be less stable. 4

25 4.3 Error estimates Again, if we assume te exact solution of (4.) is smoot, we can obtain optimal L error estimates. Suc error estimates can be obtained for te general nonlinear convection diffusion equation (4.), see [53]. However, for simplicity we will give ere te proof only for te eat equation: u t = u xx (4.5) defined on [, ] wit periodic boundary conditions. Proposition 4.3. Te solution u and q to te semi-discrete DG sceme (4.5) for te PDE (4.5) wit a smoot solution u satisfies te following error estimate (u(x, t) u (x, t)) dx + t (u x (x, τ) q (x, τ)) dxdτ C (k+) (4.6) were C depends on u and its derivatives but is independent of. Proof: Te DG sceme (4.5), wen using te notation in (4.8), can be written as B i (u,q ; v,p ) = (4.7) for all v,p V and for all i. It is easy to verify tat te exact solution u and q = u x of te PDE (4.5) also satisfies B i (u, q; v,p ) = (4.8) for all v,p V and for all i. Subtracting (4.7) from (4.8) and using te linearity of B i wit respect to its first two arguments, we obtain te error equation B i (u u,q q ; v,p ) = (4.9) for all v,p V and for all i. Recall te special projection P defined in (3.37). We also define anoter special projection Q as follows. For a given smoot function w, te projection Qw is te unique function in V wic satisfies, for eac i, (Qw(x) w(x))v (x)dx = v P k ( ); Qw(x + )=w(x i i ). (4.) I i Similar to P, we also ave, by te standard approximation teory [7], tat Qw(x) w(x) C k+ (4.) 5

26 for a smoot function w, werec is a constant depending on w and its derivatives but independent of. We now take: v = Pu u, p = Qq q (4.) in te error equation (4.9), and denote to obtain e = Pu u, ē = Qq q ; ε = u Pu, ε = q Qq (4.3) B i (e, ē ; e, ē )= B i (ε, ε ; e, ē ). (4.4) For te left and side of (4.4), we use te cell entropy inequality (see (4.)) to obtain B i (e, ē ; e, ē )= d (e ) dx + (ē ) dx + dt ˆF i+ ˆF i +Θ i (4.5) were Θ i (in fact we can easily verify, from (4.), tat Θ i = for te special case of te eat equation (4.5)). As to te rigt and side of (4.4), we first write out all te terms B i (ε, ε ; e, ē ) = (ε ) t e dx I i ε (e ) x dx +( ε ) + (e i+ ) i+ I i ( ε ) + (e i ) + i ε ē dx ε (ē ) x dx +(ε ) (ē i+ ) i+ I i (ε ) (ē i ) +. i Noticing te properties (3.37) and (4.) of te projections P and Q, weave ε (e ) x dx =, ε (ē ) x dx =, because (e ) x and (ē ) x are polynomials of degree at most k, and (ε ) i+ = u i+ (Pu) i+ =, ( ε ) + i+ = q i+ (Qq) + i+ = for all i. Terefore, te rigt and side of (4.4) becomes B i (ε, ε ; e, ē ) = (ε ) t e dx ε ē dx (4.6) ( ) ((ε ) t ) dx + (e ) dx + ( ε ) dx + (ē ) dx. 6

27 Plugging (4.5) and (4.6) into te equality (4.4), summing up over i, and using te approximation results (3.38) and (4.), we obtain d dt (e ) dx + (ē ) dx A Gronwall s inequality, te fact tat te initial error u(, ) u (, ) C k+ (e ) dx + C k+ (usually te initial condition u (, ) is taken as te L projection of te analytical initial condition u(, )), and te approximation results (3.38) and (4.) finally give us te error estimate (4.6). 4.4 Multi-dimensions Even toug we ave only discussed one dimensional cases in tis section, te algoritm and its analysis can be easily generalized to te multi-dimensional equation (4.). Te stability analysis is te same as for te one dimensional case in Section 4.. Te optimal O( k+ ) error estimates can be obtained on tensor product meses and polynomial spaces, along te same line as tat in Section 4.3. For general triangulations and piecewise polynomials of degree k, a sub-optimal error estimate of O( k ) can be obtained. We will not provide te details ere and refer to [6, 53]. 5 Discontinuous Galerkin metod for PDEs containing iger order spatial derivatives We now consider te DG metod for solving PDEs containing iger order spatial derivatives. Even toug tere are oter possible DG scemes for suc PDEs, e.g. tose designed in [6], we will only discuss te local discontinuous Galerkin (LDG) metod in tis section. 5. LDG sceme for te KdV equations We first consider PDEs containing tird spatial derivatives. Tese are usually nonlinear dispersive wave equations, for example te following general KdV type equations ( d d d u t + f i (u) xi + r i(u) g ij (r i (u) xi ) xj =, (5.) )xi i= i= 7 j=

28 were f i (u), r i (u)andg ij (q) are arbitrary (smoot) nonlinear functions. Te one-dimensional KdV equation u t +(αu + βu ) x + σu xxx =, (5.) were α, β and σ are constants, is a special case of te general class (5.). Stable LDG scemes for solving (5.) are first designed in [55]. We will concentrate our discussion for te one-dimensional case. For te one-dimensional generalized KdV type equations u t + f(u) x +(r (u)g(r(u) x ) x ) x = (5.3) were f(u), r(u) andg(q) are arbitrary (smoot) nonlinear functions, te LDG metod is based on rewriting it as te following system u t +(f(u)+r (u)p) x =, p g(q) x =, q r(u) x =. (5.4) Te finite element space is still given by (3.8). Te semi-discrete LDG sceme is defined as follows. Find u,p,q V k suc tat, for all test functions v,w,z V k and all i N, weave (u ) t (v )dx (f(u )+r (u )p )(v ) x dx +( ˆf + r ˆp) i+ (v ) ( ˆf + r ˆp) i+ i (v ) + =, (5.5) i p w dx + g(q )(w ) x dx ĝ i+ (w ) +ĝ i+ i (w ) + =, i q z dx + r(u )(z ) x dx ˆr i+ (z ) +ˆr i+ i (z ) + =. i Here again, all te at terms are te numerical fluxes, namely single valued functions defined at te cell interfaces wic typically depend on te discontinuous numerical solution from bot sides of te interface. We already know from Section 3 tat te convection flux ˆf sould be cosen as a monotone flux. It is important to design te oter fluxes suitably in order to guarantee stability of te resulting LDG sceme. In fact, te upwinding principle is still a valid guiding principle ere, since te KdV type equation (5.3) is a dispersive wave equation for wic waves are propagating wit a direction. For example, te simple linear equation u t + u xxx = wic corresponds to (5.3) wit f(u) =, r(u) = u and g(q) = q admits te following simple wave solution u(x, t) =sin(x + t), tat is, information propagates from rigt to left. Tis motivates te following coice of numerical fluxes, discovered in [55]: r = r(u+ ) r(u ) u +, ˆp = p + u, ĝ =ĝ(q,q+ ), ˆr = r(u ). (5.6) 8

29 Here, ĝ(q,q+ ) is a monotone flux for g(q), namely ĝ is a non-increasing function in te first argument and a non-decreasing function in te second argument. Te important point is again te alternating fluxes, namely ˆp and ˆr sould come from opposite sides. Tus r = r(u+ ) r(u ) u +, ˆp = p u, ĝ =ĝ(q,q+ ), ˆr = r(u+ ) would also work. Notice tat, from te tird equation in te sceme (5.5), we can solve q explicitly and locally (in cell )intermsofu, by inverting te small mass matrix inside te cell. Ten, from te second equation in te sceme (5.5), we can solve p explicitly and locally (in cell )intermsofq.tusonlyu is te global unknown and te auxiliary variables q and p can be solved in terms of u locally. Tis is te reason tat te metod is referred to as te local discontinuous Galerkin metod. 5.. Stability analysis Similar to te case for yperbolic conservation laws and convection diffusion equations, we ave te following cell entropy inequality for te LDG metod (5.5). Proposition 5.. Te solution u to te semi-discrete LDG sceme (5.5) satisfies te following cell entropy inequality d (u ) dx + dt ˆF i+ ˆF i (5.7) for some consistent entropy flux ˆF i+ = ˆF (u (x,t),p i+ (x,t),q i+ (x,t); u i+ (x +,t),p i+ (x +,t),q i+ (x + )) i+ satisfying ˆF (u, u) = F (u) +ur (u)p G(q) weref (u) = u uf (u)du and G(q) = q qg(q)dq. Proof: We introduce a sort-and notation B i (u,p,q ; v,w,z ) = (u ) t (v )dx (f(u )+r (u )p )(v ) x dx +( ˆf + r ˆp) i+ (v ) ( ˆf + r ˆp) i+ i (v ) + i (5.8) + p w dx + g(q )(w ) x dx ĝ i+ (w ) i+ I i +ĝ i (w ) + i + q z dx + r(u )(z ) x dx ˆr i+ (z ) +ˆr i+ i (z ) +. I i i 9

30 If we take v = u, w = q and z = p in te sceme (5.5), we obtain B i (u,p,q ; u,q, p ) = (u ) t (u )dx (f(u )+r (u )p )(u ) x dx +( ˆf + r ˆp) i+ (u ) ( ˆf + r ˆp) i+ i (u ) + i (5.9) + p q dx + g(q )(q ) x dx ĝ i+ (q ) i+ I i +ĝ i (q ) + i q p dx r(u )(p ) x dx +ˆr i+ (p ) ˆr i+ i (p ) + I i i =. If we denote F (u) = u f(u)du and G(q) = q g(q)dq, ten (5.9) becomes B i (u,p,q ; u,q, p )= d (u ) dx + dt ˆF i+ ˆF i +Θ i = (5.) were ˆF = F (u )+ ˆfu + G(q )+ r p + u ĝq, (5.) and Θ= ( F (u )+ ˆfu + F (u + ) ˆfu ) ( + + G(q ) ĝq G(q ) + )+ĝq+, (5.) were we ave used te definition of te numerical fluxes (5.6). Notice tat we ave omitted te subindex i in te definitions of ˆF and Θ. It is easy to verify tat te numerical entropy flux ˆF defined by (5.) is consistent wit te entropy flux F (u) + ur (u)p G(q). Te terms inside te first parentesis for Θ in (5.) is te same as tat in (3.6) for te conservation law case; tose inside te second parentesis is te same as tose inside te first parentesis, if we replace q by u, G by F,and ĝ by ˆf (recall tat ĝ is a monotone flux). We terefore readily ave Θ. Tis finises te proof of (5.7). We observe once more tat te proof does not depend on te accuracy of te sceme, namely it olds for te piecewise polynomial space (3.8) wit any degree k. Also, te same proof can be given for te multi-dimensional LDG sceme solving (5.) on any triangulation. As before, te cell entropy inequality trivially implies an L stability of te numerical solution. Proposition 5.. For periodic or compactly supported boundary conditions, te solution u to te semi-discrete LDG sceme (5.5) satisfies te following L stability d (u ) dx (5.3) dt 3