In particular, if the initial condition is positive, then the entropy solution should satisfy the following positivity-preserving property

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1 IMPLICIT POSITIVITY-PRESERVING HIGH ORDER DISCONTINUOUS GALERKIN METHODS FOR CONSERVATION LAWS TONG QIN AND CHI-WANG SHU Abstract. Positivity-preserving discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws have been etensively studied in the last several years. But nearly all the developed schemes are coupled with eplicit time discretizations. Eplicit discretizations suffer from the constraint for the Courant-Friedrichs-Levis (CFL) number. This makes eplicit methods impractical for problems involving unstructured and etremely varying meshes or long-time simulations. Instead, implicit DG schemes are often popular in practice, especially in the computational fluid dynamics (CFD) community. In this paper we develop a high-order positivity-preserving DG method with the backward Euler time discretization for conservation laws. We focus on one spatial dimension, however the result easily generalizes to multidimensional tensor product meshes and polynomial spaces. This work is based on a generalization of the positivity-preserving limiters in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 9 (1), pp ) and (X. Zhang and C.-W. Shu, Journal of Computational Physics, 9 (1), pp ) to implicit time discretizations. Both the analysis and numerical eperiments indicate that a lower bound for the CFL number is required to obtain the positivity-preserving property. The proposed scheme not only preserves the positivity of the numerical approimation without compromising the designed high-order accuracy, but also helps accelerate the convergence towards the steady-state solution and add robustness to the nonlinear solver. Numerical eperiments are provided to support these conclusions. Key words. Positivity-preserving; Discontinuous Galerkin method; Backward Euler AMS subect classifications. 65M6, 65M1 1. Introduction. In this paper, we consider the conservation law u t + f(u) =, (, t) [, π] [, + ) u(, ) = u (), [, π] (1.1) and its system version with appropriate boundary conditions. We focus on this onedimensional case, even though the result can be easily generalized to multidimensional tensor product meshes and polynomial spaces. For scalar conservation laws, it is well known that the entropy solution satisfies the following maimum principle min u () u(, t) ma u (), t. [,π] [,π] 3 31 In particular, if the initial condition is positive, then the entropy solution should satisfy the following positivity-preserving property u () = u(, t), t (1.) 3 33 For systems, even though the entropy solution does not satisfy the maimum principle in general, the physically relevant solution, for eample the density and pressure in Submitted to the editors on April 3, 17. Funding: Research supported by ARO grant W911NF and NSF grant DMS Division of Applied Mathematics, Brown University, Providence, RI 91, U.S.A. tong qin@brown.edu Division of Applied Mathematics, Brown University, Providence, RI 91, U.S.A. shu@dam.brown.edu. 1

2 TONG QIN AND CHI-WANG SHU the compressible Euler system, is always positive. In this paper, the words positive and positivity are used actually to mean nonnegative and nonnegativity. We shall use strictly positive to mean the usual positive. When designing numerical methods, we would like our numerical approimations to respect this positivity-preserving property (1.), not only because it makes the numerical approimation physically meaningful, but also it makes the numerical scheme more robust, since negative values sometimes cause ill-posedness of the problem and blow-ups of the numerical algorithm [16]. In recent years, the positivity-preserving DG schemes have been actively designed and applied for solving hyperbolic conservation laws [41, 4, 37, 38, 3, 8, 4]. All these methods are coupled with eplicit temporal discretizations, such as strong stability preserving (SSP) Runge-Kutta (RK) methods [35, 15] and multi-step methods [34]. Eplicit temporal discretizations enoy many advantages, for eample, the easiness in handling the nonlinear terms and boundary conditions, high-order accuracy with SSP properties [15], low storage requirement and so on. However, they suffer from the CFL constraint. For DG methods, to obtain the linear stability [1] or the maimum-principle stability [41], the CFL constraint becomes more and more severe as we increase the polynomial degree in the approimation space. Such stringent time stepping restriction makes eplicit methods impractical in computations involving unstructured and etremely varying meshes, viscous effect [3], low Mach numbers [3] or long-time simulations for steady-state calculation [17]. To circumvent the severe CFL constraint of eplicit methods, implicit time discretizations, which allow larger CFL numbers especially for stiff problems, are widely used in practice, especially in the CFD community to solve compressible flow problems [17, 18, 7, 3, 9, 8, 6, ] and see also the book [13]. Although most of the effort has been made for increasing accuracy of the time discretization and for increasing the efficiency of the nonlinear solver, only a few works eist in the literature concerning the positivity-preserving property of implicit methods. For compressible turbulent flow problems, Batten et al. [4] have proposed a positive finite difference scheme by splitting the flues into implicit and correction parts and the source term into positive and negative parts. The implicit and negative terms are treated implicitly via the Patankar trick [7]. In [3, 4, 5], Moryossef and Levy have developed implicit unconditional positive finite volume schemes for unsteady turbulent flows. Their main idea to preserve the positivity is to make the Jacobian matri in each implicit time step an M-matri. All these methods mentioned are low-order accurate and are complicated to generalize to high order. For DG methods, in [], Meister and Ortleb have constructed an unconditional implicit positive DG scheme for solving shallow water equations. The positivity of the numerical approimation is preserved via a modified Patankar trick [7]. The method is shown to be conservative and unconditional positivity-preserving, but only third-order accuracy is proved by a truncation error argument with no rigorous proof for arbitrary high-order spacial accuracy. In [39], Yuan, Cheng and Shu have developed a high-order unconditionally positive implicit DG method for radiative transfer equations. The positivity is preserved by utilizing the particular boundary conditions of the problem and by designing a novel rotational limiter. In this paper, we etend the general framework for constructing positivity-preserving schemes proposed by Zhang and Shu in [41, 4] to implicit temporal discretizations and develop a positivity-preserving DG method with high-order spacial accuracy for one-dimensional conservation laws. The DG methods were first introduced by Reed

3 IMPLICIT POSITIVITY-PRESERVING DG METHODS and Hill [33] for solving neutron transport equations and were further developed by Cockburn et al. in [11, 1, 9, 1] for solving the hyperbolic conservation laws. The DG method enoys mathematically provable high-order accuracy and stability. The discontinuous feature of its approimation space makes it a good fit for parallel implementation and for handling unstructured meshes. Moreover, for a class of implicit temporal discretizations, it has been shown in [19] via the cell entropy inequality that the fully discrete scheme for the nonlinear conservation law is unconditionally L -stable, which works for arbitrary triangulation and any spacial order of accuracy. We adopt the backward Euler temporal discretization in this paper. Our focus is on constructing a spatially high-order positivity-preserving DG scheme. The main conclusion is that in order to generalize the Zhang-Shu positivity-preserving limiter [41, 4] to the backward Euler DG scheme, a lower bound for the CFL number is required. This is proved theoretically for linear scalar equations and numerically verified for nonlinear equations. The proposed positivity-preserving limiter is inepensive and easy to implement. It not only preserves the positivity and high-order spacial accuracy but also makes the numerical scheme more robust, in the sense that it accelerates the convergence towards the steady-state solution and adds robustness to the nonlinear solver for etreme test cases. The organization of the paper is as follows. In Section we describe the DG scheme. Then in Section 3 the positivity-preserving technique is introduced for scalar equations. In particular, a CFL condition is derived for linear equations to ensure the positiveness of the scheme. The positivity-preserving DG scheme for the compressible Euler system follows in Section 4. Numerical eperiments are presented in Section 5 and concluding remarks are given in Section 6.. Implicit DG scheme..1. The DG discretization. In this section, we define the DG scheme for (1.1). First, let us fi some notations. We decompose the domain Ω = [, π] into N subintervals, I = [ 1, + 1 ], for = 1,,, N. The size of each subinterval is denoted by h. Define Î = [ 1, 1] to be the reference cell and define T () = ( )/h to be the affine mapping between the intervals I and Î, where = ( )/ is the midpoint of I. Moreover, let (, ) denote the usual L inner product on I and (, )Î the one on Î. Then we define the approimation space V h = {v L (Ω) : v I P k (I ), = 1, N} where P k (I ) denotes the polynomial space on I with degree up to k. The semi-discrete DG scheme is to seek the approimation u h (t) V h, such that in each subinterval I, d dt (u h(t), v) (f(u h (t)), v ) + ˆf + 1 (u h(t))v( ) ˆf (u h(t))v( + ) = (.1) holds for any v V h, where ˆf + 1 (u) = ˆf(u( u( + ), + )) and ˆf(, ) is the numerical flu function. In this paper, we consider the global La-Friedrichs flu ˆf(a, b) = 1 [f(a) + f(b) α(b a)] (.) 113 where α = ma Ω f (u ()).

4 4 TONG QIN AND CHI-WANG SHU Time discretization. With shorthand notation, the semidiscrete scheme (.1) can be rewritten as below d dt (u h(t), v) = L (u h (t), v), v V h, = 1,..., N (.3) [ where L (u h (t), v) = (f(u h (t)), v ) ˆf+ 1 (u h(t))v( ) ˆf ] (u h(t))v( + ). 1 We use the backward Euler method to further discretize this ODE system. Then the fully discrete scheme is defined by seeking the approimation at time t n+1, which is denoted by u n+1 h V h, such that in each cell I, we have (u n+1 h, v) tl (u n+1 h, v) = (u n h, v) (.4) for all v V h. In the following, we use u n to denote un h I, (u n ) ± to denote u n + 1 (± ) + 1 and use ū n to denote the cell average of un in the interval I. To further solve the nonlinear system (.4), there have been many works on how to build efficient solvers, such as the work in [3, 9, 8]. But since our main focus is on the positivity preserving property rather than the efficiency of the nonlinear solver, we use the Newton method [13] for the nonlinear system up to accuracy For the robustness and accuracy reasons, in each Newton iteration, the Jacobian matri is solved with the direct solver. 3. Positivity-preserving DG scheme for scalar equations. In this section, we introduce how to add the positivity-preserving property to the scheme (.4). First, let us give the definition of the positivity-preserving DG scheme for the scalar equation as that in [41]. Definition 3.1. A DG scheme is defined to be positivity preserving if given u n h () 13, for any Ω, then we have u n h (), Ω. 134 This definition will be slightly modified later (requiring positivity on specified quadra- 135 ture points rather than on all points) in order to obtain more efficient implementation. 136 Generally, the original high-order DG method is not positivity-preserving. We follow 137 the general approach in [41] and construct high-order positivity-preserving DG meth- 138 ods in the following two steps. 139 Step 1 First, in each cell I, given u n () for any I, find a sufficient condition 14 such that we have part of the approimation at the net time level u n+1 (), 141 i.e., the cell average ū n+1 positive. Step Net, we make the whole polynomial u n+1 14 () by invoking the scaling 143 limiter in [1, 41]. 144 The main difficulty lies in the first step. Unlike the eplicit scheme, for implicit DG methods, the polynomial u n+1 () depends on the previous information u n h () in a 146 global and implicit way. In this section, we would first show how to overcome this 147 difficulty for scalar linear equations and derive a CFL condition, under which, given u n 148 () for all, we have the cell average ūn+1 is positive for all. Then we will 149 introduce the scaling limiter and summarize the algorithm Preliminaries. Let us first recall some definitions and results that will be useful in the following analysis. The first useful tool is the so-called M-matri. For a thorough introduction, one can refer to [5]. To define it let us first set Z n n = {A = (a i ) R n n : a i, i }

5 IMPLICIT POSITIVITY-PRESERVING DG METHODS 5 which is the set of all the n n real matrices with nonpositive off-diagonal entries. In [31], the author listed forty equivalent characterizations for M-matrices. For our purpose, we adopt the following one as the definition. Definition 3.. A matri A Z n n is called an M-matri if A is inverse-positive, that is, A 1 eists and each entry of A 1 is nonnegative. M-matrices have the following equivalent characterization [31]. Theorem 3.3. A matri A = (a i ) Z n n is an M-matri if and only if a ii >, 1 i n, and there eists a positive diagonal matri D = diag{d 1,, d n } such that AD is strictly diagonally dominant, that is, a ii d i > i a i d for 1 i n. In particular, if D is the identity matri, we have the following corollary. Corollary 3.4. A matri A = (a i ) Z n n is an M-matri if a ii > and it is strictly diagonally dominant. In the following, we also utilize properties of Legendre polynomials. We consider the standard Legendre polynomials {p n ()} defined on Î by the following recursive 166 relationship 167 (n+1)p n+1 () = (n+1)p n () np n 1 (), p () = 1, p 1 () =, Î (3.1) In the following lemma, we collect some properties of Legendre polynomials that will be useful in the following analysis. For the proof, one can refer to [6]. Lemma 3.5. Legendre polynomials defined in (3.1) have the following properties 1. p n (1) = 1, p n ( 1) = ( 1) n.. Î p n()p m ()d = n+1 δ nm, where δ nm is the Kronecker delta. 3. (n + 1)p n () = d d [p n+1() p n 1 ()]. 4. Rodrigues formula p n () = 1 d n n n! d [( 1) n ]. n 3.. CFL condition for positivity-preserving for linear equations. We consider the linear equation u t + u =, u(, ) = u (), Ω (3.) 177 with periodic boundary condition. Then the scheme (.4) becomes (u n+1 h, v) t(u n+1 h, v ) + t[(u n+1 ) v (u n+1 ) v + ] = (u n h, v) (3.3) Given u n 178 h (), Ω, we want to derive a CFL condition, under which the cell average ū n+1,. In order to do that, we first epress ū n+1 in terms of u n 179 h. The 18 idea is to take k + 1 different test functions as probes to etract the information out from u n+1 h () in terms of u n 181 h (). First, let us take v = 1 in the scheme (3.3), 18 we have ū n+1 + λ [(u n ) (u n+1 ) ] = ū n 1, = 1,, N 183 where λ = t h. We can rewrite the system above in the matri form as below Λ 1 ū n+1 + A(u n+1 ) = Λ 1 ū n (3.4)

6 6 TONG QIN AND CHI-WANG SHU where ū n = (ū n 1,, ūn N )T, (u n+1 ) = ((u n+1 3 ),, (u n+1 ) ), Λ and A are N N N+ 1 matrices in the following form λ 1. Λ = λ , A = λ N Net, we need to epress the cell boundary value (u n+1 ) by u n h. To this end, we need to take other special test functions. Recall that the Dirac delta function has the following epansion δ( y) = 1 (l + 1)p l ()p l (y),, y Î l= where p l () is the standard Legendre polynomial defined in (3.1). Then we set y = 1, truncate the summation at the (k + 1)th term and define ˆδ k () = 1 k l= (l + 1)p l () = k + 1 p k+1 () p k (), 1 Î (3.5) 191 We have employed the Christoffel-Darbou formula [14] in the last equality. 19 The following lemma says that this polynomial actually defines the Dirac delta k function in P (Î) at the point y = Lemma 3.6. The polynomial ˆδ k has the following properties ˆδ k () P k (Î) and for any w P k (Î), we have (w, ˆδ k )Î = w(1).. In the cell I, define δ k () = h ˆδk (T ()), I (3.6) then it is the delta function in P k (I ) at The mass is concentrated at = 1, in the sense that for any k and =,, k 1, we have (ˆδ k ) () (1) (ˆδ k ) () 199 () > for any [ 1, 1). Proof. It is obvious that ˆδ k P k (Î). For any polynomial w Pk(Î), we can write 1 it as a linear combination of Legendre polynomials as below k w() = c l p l (), l= Î Then by the definition of ˆδ k and Lemma 3.5, we have (w, ˆδ k k )Î = c l (p l, ˆδ k k k )Î = c l = c l p l (1) = w(1) l= l= l= 3 The second property can be shown by a simple change of variable.

7 IMPLICIT POSITIVITY-PRESERVING DG METHODS For the third property, by the definition of ˆδ k, it suffices to show p (i) l (1) p (i) l () > (3.7) for l =, 1,, k 1, i =,...,l 1 and for any [ 1, 1). It is straightforward to verify that (3.7) holds for l =, 1,. For l >, by Lemma 3.5, we have, p (i) l () = (l 1)p (i 1) l 1 () + p(i) l () If (3.7) holds for l m 1, then we have when l = m, for fied i < m and any [ 1, 1), p (i) m (1) = (m 1)p(i 1) m 1 (1) + p(i) m (1) > (m 1)p(i 1) () + p(i) m 1 m () = p(i) m () Then by induction, we have proved (3.7) and hence the third property in the lemma. With the help of the discrete delta function (3.6), we have the following representation of (u n+1 + ). 1 Lemma 3.7. For linear scalar conservation law with f(u) = u discretized by the scheme (.4), we have σ k (u n+1 ) = ξ + 1 k ū n+1 + (û n, g k )Î (3.8) where and k 1 σ k = 1 + (λ ) i+1 (α k i βi k ), ξ k = i= k 1 [ ] g k () = (λ ) i (ˆδ k ) (i) () βi k i= k (λ ) i βi k, α k i = (ˆδ k ) (i) (1), β k i = (ˆδ k ) (i) ( 1) (3.9) i= Proof. For fied l =, 1,, k 1, take the test function to be (δ k)(l) () (δ k)(l) ( 1 ) in the scheme (3.3). By the definition of δk () and Lemma 3.6, we have (u n+1, (ˆδ k ) (l) (T ())) h βl k ūn+1 λ (u n+1, (ˆδ k ) (l+1) (T ())) + t(u n ) (α k l β k l ) = (u n, (ˆδ k ) (l) (T ())) h β k l ū n 19 If we epand u n+1 in terms of the basis φ l () = p l(t ()), l =,...,k, as u n+1 k l= (cl )n+1 φ l and by a change of variable, we obtain = (û n+1, (ˆδ k ) (l) )Î β k l ū n+1 λ (û n+1, (ˆδ k ) (l+1) )Î + λ (u n ) (α k l β k l ) = (û n, (ˆδ k ) (l) )Î β k l ūn 1 where û n = k l= (cl )n p l (). Or equivalently,

8 8 TONG QIN AND CHI-WANG SHU (û n+1, (ˆδ k ) (l) )Î λ (û n+1, (ˆδ k ) (l+1) )Î = β k l ūn+1 λ (u n ) (α k l βk l ) + (ûn, (ˆδ k ) (l) )Î β k l ūn If we set D l = (û n+1, (ˆδ k ) (l) )Î, C l = β k l ū n+1 λ (u n ) (α k l β k l )+(û n, (ˆδ k ) (l) )Î β k l ū n 3 then we have D l λ D l+1 = C l, l =,, k 1 4 and in particular, when l = k 1, we have D k 1 = λ D k + C k 1 = λ (û n+1, (ˆδ k ) (k) )Î + C k 1 = 4λ βkū k n+1 + C k 1 5 Then we have the following derivations (u n+1 ) = (u n+1 + 1, δ k ) = (û n+1, ˆδ k )Î = D = λ D 1 + C l 1 = (λ ) D + λ C 1 + C = = (λ ) l D l + C i (λ ) i = k = (λ ) k 1 [4λ βkū k n+1 + C k 1 ] + (λ ) i C i = (λ ) k β k kū n+1 k 1 + (λ ) i C i i= i= i= 6 Plugging in the definition of C i and after some manipulation, we obtain [ ] k (λ ) i+1 (α k i βi k ) (u n+1 ) = i= [ ] k βi k (λ ) i i= k 1 + ū n+1 (λ ) i (û n, (ˆδ k ) (i) βi k )Î i= 7 For the parameters {β k 8, αk }N =1, we have the following lemma, which will be useful in the proof of Proposition 3.1. The proof will be provided in Appendi A. 9 3 Lemma 3.8. For any k, the following results hold 1. α k i > βk i for i =,, k 1 and hence σ k 31 > for any.. βk i k 3 >, βk k i 1 <, for i =,, k/. 3. βk i k 33 + βk k i 1 >, for i =,, k/. With Lemma 3.7 and the equation (3.4), we can obtain the following cell average equation. Tū n+1 = L(u n ) (3.1)

9 IMPLICIT POSITIVITY-PRESERVING DG METHODS 9 36 where T = ξ k 1 σ k λ 1 ξk N σ k N ξk 1 σ k 1. ξ k σ k λ ξk N 1 σ k N 1 ξn k + 1 σn k λ N and (L(u n )) = ( û n, 1 λ gk σ k ) Î + ( û n 1, gk 1 σ k 1 ) Î, = 1,...,N. A set of sufficient conditions to make the cell average ū n+1 positive for any = 1,, N are the following Condition I T is an M-matri, or by Corollary 3.4 and Lemma 3.8, ξ k >, = 1,, N (3.11) 41 Condition II L(u n ) is positive, or (û n, 1/λ g k /σk ) Î + (ûn 1, gk 1 /σk 1 ) Î >, = 1,, N (3.1) By Lemma 3.6, it is easy to obtain g k() < σk λ, Î, for any k and and hence 4 the first term in (3.1) is always positive. Since û n and ûn 1 can be any independent 43 positive polynomials, the condition (3.1) is further reduced to 44 (v, g k ) k Î, v P (Î) and v (3.13) In summary, if we set F k (λ, ) = k i= (λ)i (ˆδ k ) (i) () and then ξ k = F k (λ, 1), g k = F k(λ, ) F k (λ, 1), sufficient conditions (3.11) and (3.1) actually require that the CFL number λ satisfy F k (λ, 1) (3.14) k (F k (λ, ) F k (λ, 1), v( ))Î, v P (Î) and v. (3.15) The following theorem says that in order to make these two conditions hold simultaneously, the CFL number λ can not be arbitrarily small. 5 Theorem 3.9. When λ is small, conditions (3.14) and (3.15) can not hold at the 51 same time. More specifically, we have the following two cases: 1. When the polynomial degree k is odd, there eists η1 k 5 > such that when λ < η1 k 53, the condition (3.14) does not hold.. When k is even, there eists η k > such that when λ < η k 54, the second 55 condition (3.15) does not hold Proof. When k is odd, F k (λ, 1) = k i= βk i (λ ) i is a polynomial of odd degree. By Lemma 3.8, we have the leading coefficient βk k > and hence F k(λ, 1) > when λ is large. On the other hand, when λ =, F k (, 1) = β k <, again by Lemma 3.8. Therefore, the polynomial F k (λ, 1) must have at least one and at most k positive roots. If we take η1 k to be the smallest one, we would have the first statement.

10 1 TONG QIN AND CHI-WANG SHU 61 When k is even, in (3.15) take v = 1 and we obtain (F k (λ, ) F k (λ, 1), 1)Î = k 1 [ ] (λ ) i (ˆδ k ) (i 1) (1) (ˆδ k ) (i 1) ( 1) (ˆδ k ) (i) ( 1) i= where (ˆδ k ) ( 1) (1) (ˆδ k ) ( 1) ( 1) := ˆδ k ()d. To simplify the notation, let us set Î y = λ and set k 1 [ ] G k (y) = y i (ˆδ k ) (i 1) (1) (ˆδ k ) (i 1) ( 1) (ˆδ k ) (i) ( 1). i= Since k 1 is odd, again, we would like to show it has positive real roots. First, when y =, we have G k () = ˆδ k ()d ˆδ k ( 1) = 1 k + 1 = 1 k 1 Î 66 Net, let us check the leading coefficient of G k (y). Since (ˆδ k ) (k) = βk k, we have 67 (ˆδ k ) (k ) = 1 βk k +C 1 +C, where C 1 and C are constants. As a consequence, we 68 have α k k = 1 βk k + C 1 + C, β k k = 1 βk k C 1 + C, β k k 1 = β k k + C 1 69 and hence the leading coefficient of G k (y) satisfies (ˆδ k ) (k ) (1) (ˆδ k ) (k ) ( 1) (ˆδ k ) (k 1) ( 1) = α k k β k k β k k 1 = β k k > 7 Therefore, the odd-degree polynomial G k (y) must have at least one and at most k 1 71 positive real roots. If we take η k to be the smallest one, we can conclude the second 7 statement. 73 Theorem 3.9 indicates that unlike the situation for the DG method with Euler 74 forward time discretization in [41], where an upper bound for the CFL number is 75 sufficient for the cell average at the net time level to be positive, for the DG scheme 76 with the Euler backward time discretization, the CFL number can not be arbitrarily 77 small and an lower bound may be required. The following analysis confirms this 78 statement. 79 If we re-eamine the condition (3.15) and note that for fied λ, F k (λ, ) k P (Î), the inner product in (3.15) can actually be approimated eactly by certain quadrature rules, say {( α, ω α )} Nq 81 α=1, where {α } are the abscissas in Î, {ωα } the 8 weights and N q is large enough such that the quadrature rule is eact for polynomials of degree k. We denote {( α, 83 ωα )}Nq α=1 to be the transformed quadrature rule in I. 84 Then the condition (3.15) can be reduced to require J k α (λ ) := F k (λ, α ) F k (λ, 1), α = 1,, N q (3.16) If we define J k(λ 85 ) = F k (λ, 1), together with condition (3.14), we require the CFL number to make N q +1 polynomials {Jα(λ k )} Nq α= positive. The following result states 86 that it suffices to require λ 1. 87

11 IMPLICIT POSITIVITY-PRESERVING DG METHODS Proposition 3.1. When min λ 1, we have Fk(λ, 1) >, (3.17) F k (λ, ) F k (λ, 1) >, Î (3.18) 89 9 Proof. First, let us consider (3.17). By the definition we have F k (λ, 1) = k i= (λ ) k βi k, which can be rewritten as F k (λ, 1) = (λ ) k 1 (λ β k k + β k k 1) + + { (λ β k + βk 1 ) + βk, if k is even (λ β1 k + β), k if k is odd (3.19) When λ > 1/, i.e., λ > 1, then by Lemma 3.8 it is easy to see that each term in (3.19) is strictly positive and hence F k (λ, 1) >. For the condition (3.18), consider the ith derivative of F k with respect to. i i F k(λ, ) = k (λ ) l i ( δ ˆk ) (l) (), l=i i =,...,k When i = k, by Lemma 3.6, we have k F k k (λ, ) = βk k > and hence k 1 k 1 F k(λ, ) > k 1 k 1 F k(λ, 1) = βk 1 k + (λ )βk k >, Î. We have used the facts λ 1 and Lemma 3.8 to derive the last inequality. We can continue this procedure till i = and obtain F k (λ, ) > F k (λ, 1), Î Therefore, in summary, when min λ 1, we have conditions (3.17) and (3.18) hold and hence we can draw the conclusion. Consequently, when λ 1 all the polynomials {Jk α} k 99 α= are strictly positive. On 3 the other hand by the proof of Theorem 3.9, at λ = these polynomials can not be 31 all nonnegative. This implies that if we denote R k to be the set of all the positive roots of each polynomial Jα k, i.e., 3 Rk = {r R + : Jα k (r) = for some α =,, N q} 33 then we must have R k, R k (, 1 ) and R k is finite. Then if we set we have the following theorem. r k = ma r R k r, (3.) Theorem For the backward Euler DG scheme (3.3) with P k element for linear equation (3.), at time level n given u n (α ) on the quadrature points {α }Nq α=1 I which give eact approimation for polynomials of degree k, we have ū n+1, for any under the following CFL condition min λ r k (3.1) 39 where r k (, 1 ) is defined as in (3.).

12 TONG QIN AND CHI-WANG SHU Proof. When λ r k by the definition of R k, we have (3.16) and (3.14) hold. Therefore T in (3.1) is an M-matri and L. By the definition of M-matri, we can conclude the result. Remark 3.1. We only require u n 313 h to be positive on quadrature points {α }Nq α=1, which is weaker than the condition u n h (), for any Ω in the Definition Remark Even though the Theorem 3.11 is only proved for linear equations, numerical eperiments suggest that for nonlinear equations, a lower bound for the CFL number is still necessary to make the cell average at the net time level positive Remark The results also hold for problems with positive source terms and positive inflow boundary conditions Remark The lower bound depends on the polynomial degree k as well as the 31 quadrature rule we choose. For each k and fied quadrature rule we can actually 3 obtain the lower bound r k by solving the positive roots of each Jα. k In Table 3.1, we 33 record the lower bounds for Legendre-Gauss-Lobatto (LGL) quadrature rule and the 34 Legendre-Gauss (LG) rule respectively. We see that the LGL rule gives smaller lower bound. In practice we will limit the polynomial u n to make it positive at least on the 36 LGL points { α }Nq α=1 in each cell I. Table 3.1 Values of r k for k = 1,,5 and for Legendre-Gauss-Lobatto (LGL) and Legendre-Gauss quadrature rules respectively. LGL rule LG rule k N q r k N q r k Remark The lower bounds in Table 3.1 are sharp for odd k and sufficient for even k. If we start from the following initial condition, u () = { 1, if I M, otherwise, where M = argma h. After one step we record the minimum cell averages for different odd k and λ in Table 3.. We see that when λ is slightly smaller than the lower bound r k, after one time step, at least one of the cell averages will become negative. If λ is larger than r k, the average will be uniformly positive. For even k, we consider a different initial condition ( k ( M) u () = h M.7), if IM, otherwise

13 IMPLICIT POSITIVITY-PRESERVING DG METHODS 13 Table 3. Minimum cell average after one time step for odd k. k λ = r k.1 min ū 1 λ = r k +.1 min ū E E E E E E The Table 3.1 shows the minimum cell averages for different k and λ. We see that the lower bound for the CFL number is still necessary for the positivity of ū 1 and r k listed in Table 3.1 is sufficient. Table 3.3 Minimum cell average after one time step for even k. k λ min ū 1 λ = r k min ū E E E E Scaling limiter. Once in each cell I, the cell average ū n+1 is positive, we limit the whole polynomial u n+1 () towards its cell average by utilizing the following scaling limiter [1, 41]. ũ n+1 = θ [u n+1 ū n+1 ] + ū n+1 (3.) 34 where ū n+1 θ =, if min ū n+1 min I u n+1 () I u n+1 () < 1, otherwise (3.3) This procedure preserves the original high-order accuracy [41]. Lemma For the modified polynomial ũ n+1 (), we have ũ n+1 () u n+1 () C k ma u n+1 I () u() where u is the smooth solution. Basically, what this lemma says is that the error we commit in the limiting procedure is bounded by the error of the original approimation up to a constant depending on k. The proof can be found in [4]. Remark In order to calculate the scaling parameter θ in (3.3) we need to calculate the minimum value of u n+1 in each cell I, which can be done efficiently up to k = 5 via the root formulas. For larger k, this calculation becomes epensive. But recall that in Theorem 3.11, we only require each polynomial u n+1 to be positive at the LGL quadrature points { α }Nq α=1 and hence we can instead use the following

14 14 TONG QIN AND CHI-WANG SHU 35 scaling parameter ū n+1 θ = ū n+1 min α u n+1 ( α ), if min α u n+1 ( α ) < 1, otherwise (3.4) in the limiter. This is similar with the scaling limiter in [41] and the same argument can be conducted here to prove that the modified limiter does not kill the original high-order accuracy either Algorithm for scalar equations. Now, we can summarize the positivity- 357 preserving algorithm for scalar equations as below. 1. At time level t n, given u n 358 () being positive at least on the LGL quadrature points { α 359 }Nq α= Choose CFL number large enough a priori or adaptively enlarge it in each time step until ū n+1 361,. 3. Apply the scaling limiter (3.) with θ defined in (3.3) or (3.4) to u n+1 36 such that u n+1 ( α ) ǫ, for any α = 1,...,N 363 q and, where ǫ is a small 364 number to help get rid of the round-off effect. In the numerical eamples, we take ǫ = Positivity-preserving DG scheme for compressible Euler systems. We consider the following compressible Euler system for ideal gas u t + f(u) =, t, [, 1]. (4.1) 368 with u = (ρ, m, E) T, f = (m, ρv + p, (E + p)v) T and m = ρv, E = ρv + ρe, p = (γ 1)ρe, where ρ is the density, v the velocity, 37 m the momentum, p the pressure, E the total energy and e is the internal energy. The constant γ > 1 is the ratio of specific heats. The Jacobian matri f u has the 371 following three eigenvalues ζ 1 = v c, ζ = v, ζ 3 = v + c, where c = γp/ρ is the sound speed. 374 The physical solution lies in the following admissible set { ( G = u = (ρ, m, E) T : ρ, p = (γ 1) E 1 m ) } (4.) ρ It can be verified that G is a conve set [4]. The DG scheme for (4.1) is to seek approimation vector u h (t) V h = [V h ] 3 such that in each cell I we have d dt (u h(t),v) = L (u h (t),v), v V h, (4.3) where L (u h (t),v) = (f(u h (t)),v ) [ˆf+ 1 (u h(t))v( ) ˆf (u h(t))v( + ) 1 The numerical flu ˆf(, ) is taken to be the global La-Friedrichs flu ]. 1 ˆf(a,b) = [f(a) + f(b) α(b a)]

15 IMPLICIT POSITIVITY-PRESERVING DG METHODS with α = c + v. 381 As for the scalar case, the ODE system (4.3) is solved by the backward Euler method. If we let u n h to denote the DG approimation at time level n, then the 383 positivity-preserving DG scheme for compressible Euler system is defined as below. 384 Definition 4.1. A DG scheme for compressible Euler system is positivity-preserving if at time level n given u n 385 h () G for all Ω, then at the net time level (n + 1), we have u n h () G for all Ω. 387 We design the positivity-preserving DG scheme by etending the positivity-pres- 388 erving limiter in [4] and the more robust version in [36] for the eplicit time stepping 389 to the backward Euler time stepping. We stress on the applicability of the proposed 39 method rather than its theoretical ustification. The analysis for the linear scalar equation suggests that starting from u n 391 () G, in order to have the cell average ū n+1 39 G, a lower bound for the CFL number may be required. Therefore, we 393 formulate the algorithm as follows. 1. At time level t n, in each cell I, given u n 394 (α ) G at the LGL quadrature points { α }Nq α= Choose a large enough CFL number a priori or adaptively enlarge it in each time step until ū n G for any In each cell I, apply the following scaling limiter to the first component of u n+1 to obtain ρ n+1 () at the LGL quadrature points { α 399 }Nq α=1 4 where ρ n+1 = θ 1 (ρ n+1 ρ n+1 ) + ρ n+1 ρ n+1 θ 1 = ρ n+1 min α ρ n+1 ( α ), if min α ρ n+1 ( α ) < 1, otherwise Denote the modified polynomial by ũ n In each cell I, apply the scaling limiter again to the whole modified polynomial ũ n+1 such that p( α ) for each α as below. 45 where ũ n+1 = θ (ũ n+1 ũ n+1 ) + ũ n+1 p n+1 θ = p n+1 min α p n+1 ( α ), if min α p n+1 ( α ) < 1, otherwise and the pressure average is defined by p n+1 = p( ũ n+1 ). 5. Numerical eperiments. In this section, we present numerical eamples. First, we verify the high-order spacial accuracy of the proposed method by testing it on both linear and nonlinear steady-state problems. An acceleration of the convergence towards the steady state solution is also observed. Net, we test the methods on moving-shock problems. At last, eamples for compressible Euler system will be presented. In all the eamples, the domain is first uniformly decomposed and then each node is randomly perturbed up to % of the mesh size.

16 16 TONG QIN AND CHI-WANG SHU Accuracy tests. First, let us test the accuracy of the proposed method. We check the spacial accuracy with the steady-state solution to both of the linear equation and the Burgers equation. We take t = 1 ma h and march in time until u n+1 h u n h 1 1. Eample 5.1 (Steady-state solution to linear problem). For the linear equation, we consider the following problem u t + u = sin 4 (), u(, ) = sin (), u(, t) = (5.1) with the outflow boundary condition at = π. The eact solution u(, t) can be derived by the characteristic theory and can be shown to be positive for all t >. In Table 5.1 and Table 5. we record the errors, numerical orders of accuracy and the minimum value of the numerical approimation, with and without the positivitypreserving limiter respectively. We see that without the positivity-preserving limiter the minimum value of the steady-state approimation is negative. When the limiter is put on, the minimum value becomes positive and the high-order accuracy is not destroyed. Table 5.1 Error table for Eample 5.1, approimation of the steady state solution to the linear problem (5.1), without the positivity preserving limiter. k N L error order L error order min u h E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Eample 5. (Steady-state solution to Burgers problems). For the Burgers equation, we consider the steady state solution to the following problem ( ) u ( u t + = sin, u(, ) =, u(, t) = (5.) 4)

17 IMPLICIT POSITIVITY-PRESERVING DG METHODS 17 Table 5. Error table for Eample 5.1, approimation of the steady state solution to the linear problem (5.1) with the positivity preserving limiter. k N L error order L error order min u h 4.53 E E- 1. E E E E E E E E E E E E E E E-3 1. E E E E E E E E E E E E E E-3.4 E-3 1. E E E E E E E E E E E E E E E-5 1. E E E E E E E E E E E E E with the outflow boundary condition at = π. Again, by the characteristic theory, one can show that the solution to this problem is always positive. In Tables 5.3 and 5.4, we present numerical results for the cases where the positivity-preserving limiter is on and off respectively. This eample shows the effectiveness of the positivity preserving limiter for the nonlinear scalar problem. With the limiter on, the solution stays positive and the high-order accuracy is preserved. Net, let us turn to another steady-state Burgers problem ( ) u ( u t + = sin 3 ) (, u(, ) = sin ), u(, t) = (5.3) 4 4 with the outflow boundary condition at = π. This problem is more difficult than the previous one, since the source is much closer to zero around =. In Table 5.5, we present the error, numerical convergence rate as well as the number of time steps taken to reach the steady state solution, which is denoted by N T. Both cases where the positivity preserving limiter is on and off are presented respectively. We use this eample to illustrate that the stability added to the scheme by the positivitypreserving limiter helps accelerate the convergence towards the steady-state solution. 5.. Moving Shocks. Net, we test the proposed scheme on problems involving moving shocks. In all the numerical eperiments below, quadratic P element and non-uniform mesh are employed.

18 18 TONG QIN AND CHI-WANG SHU Table 5.3 Error table for Eample 5., approimation of the steady state solution to the Burgers equation (5.), without the positivity preserving limiter. k N L error order L error order min u h.915 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-1 Table 5.4 Error table for Eample 5., approimation of the steady state solution to the Burgers equation (5.), with the positivity preserving limiter. k N L error order L error order min u h 3.7 E E-6 1. E E E E E E E E E E E E-9 1. E E E E E E E E E E E E-1 1. E E E E E E E Eample 5.3 (Linear Problem). The first eample is the linear equation with the initial data { 1, if [3, 4] u () =, otherwise In Figure 5.1, we present the numerical results at T =., with and without the positivity-preserving limiter respectively. For the linear equation, we take min λ = r =.66 as listed in Table 3.1. As the zoom-in plots show, the limiter helps the solution to stay positive. Without the limiter a negative undershoot appears. Eample 5.4 (Burgers problem). Net, let us consider the Burgers equation with the initial condition u () = 1 + sin() and periodic boundary conditions. The initial condition is positive and takes value zero at = π. At T = 1.5, a shock is developed. In Figure 5., we show the numerical approimation with and without the limiter

19 IMPLICIT POSITIVITY-PRESERVING DG METHODS 19 Table 5.5 Error table for Eample 5., approimation of the steady-state solution to the Burgers equation (5.3), with and without the positivity preserving limiter. without limiter with limiter k N L error order N T L error order N T 1.71 E E E E E E E E E E E E E E u u (a) without limiter (b) zoom in around = u u (c) with limiter (d) zoom in around = Fig Eample 5.3: at T =., with t =.66 ma h and N = 1. Solid line is the eact solution. Blue squares are point values at the Legendre-Gauss-Lobatto points respectively. We see that even though the profiles are smeared due to the first-order accuracy of the backward Euler time discretization, the effectiveness of the positivitypreserving limiter can still be observed in the zoom-in plots around the shock. Eample 5.5 (Buckley-Leverett problem). In this eample, the Buckley-Leverett prob-

20 TONG QIN AND CHI-WANG SHU u 1 u (a) without limiter (b) zoom in around the shock u 1 u (c) with limiter (d) zoom in around the shock Fig. 5.. Eample 5.4 at T = 1.5. Solid line is the eact solution. Blue squares are point values of the numerical approimation on the Legendre-Gauss-Lobatto points. Mesh with N = 1 cells and CFL = u lem is considered, with f(u) = 4u +(1 u). For this flu function, we have f () = f (1) =. If we start with the usual step function u () = I [3,4], numerical eperiments indicate that no matter how large t is, we can not make the cell average ū 1 positive for all. The same phenomena is also observed for the Burgers problem with step function as the initial condition. This indicates that the lower bound for the CFL number is also necessary for the positivity-preserving limiter to work for nonlinear problems, and in general, the limiter may not be an effective positivity-preserving tool when applied to problems with sonic points. In this eample, we change the initial condition to {.9, if [3, 4] u () = 1 3, otherwise In Figure 5.3, we present the plots with and without limiter. Even though the profile is smeared around the shocks and the rarefaction wave by the first-order timediscretization, the positivity-preserving property is observed in the zoom-in plots Compressible Euler System. At last, we turn to the eamination of the applicability of the proposed scheme to the compressible Euler system (4.1). The

21 IMPLICIT POSITIVITY-PRESERVING DG METHODS u.4 u (a) without limiter (b) zoom in around = u.4 u (c) with limiter (d) zoom in around = 3 Fig Eample 5.5 at T =.6. Solid line is the eact solution. Blue squares are point values of the numerical approimation on the Legendre-Gauss-Lobatto points. Mesh with N = 1 cells and CFL = computational domain Ω = [, 1] is decomposed into nonuniform mesh. The generic ratio of specific heats is taken to be γ = 1.4. For all the eamples, quadratic element is employed with the positivity-preserving limiter on. And the time stepping size is set to be t = CFL c+ v min h, where c is the sound speed and v is the velocity. We take CFL = for all the eamples. Eample 5.6 (Shock tube). First, let us consider the following shock tube problem. ρ = 1, if <.5 v =, if <.5, p = 1, if <.5 ρ = 1, if.5 v =, if.5 p =.1, if The numerical approimation is presented in Figure 5.4. The solutions consists of a strong shock wave, a contact discontinuity and a rarefaction wave. All these features are well captured by our scheme. Moreover, with the positivity-preserving limiter on, both the pressure and the density stays positive during the simulation. Eample 5.7 (Double rarefaction). The double rarefaction problem starts from the

22 TONG QIN AND CHI-WANG SHU p (a) density (b) pressure Fig Eample 5.6 at T =.1 with N = and CFL =. With the positivity-preserving limiter on. Solid line is the eact solution and blue squares are numerical approimations. 488 following initial condition ρ = 1, if <.5 v =, if <.5, p =.4, if <.5 ρ = 1, if.5 v =, if.5 p =.4, if This problem has solution consisting of two symmetric rarefaction waves and trivial contact wave of zero speed. The region between the nonlinear waves around =.5 is close to vacuum, which brings difficulty to the simulation. Without the limiter, the nonlinear solver will eperience a hard time to converge. In Figure 5.5, we show the plots for the pressure and the density and we see that the near-vacuum region is well resolved with the help of the positivity-preserving limiter p (a) density (b) Pressure Fig Eample 5.7 at T =.1, with N = and CFL =. With the positivity-preserving limiter on. Solid line is the eact solution and blue squares are the numerical approimations Eample 5.8 (Blast wave). The last eample is the Sedov point-blast wave []. Initially the gas is steady with uniform density one in the whole domain. The pressure is set to be p = 1 9, ecept in the central cell, where the pressure is as high as p = 1 4.

23 IMPLICIT POSITIVITY-PRESERVING DG METHODS Then a blast-wave starts to propagate from the central cell with a shock front. This problem is difficult, since it involves a low density region and strong shocks. In Figure 5.6 we present the numerical approimation and the eact solution [] for the pressure and density respectively. The positivity-preserving limiter not only helps keep the low-density region positive, but also add robustness to the nonlinear solver. Without it, the code breaks down due to the failure of convergence of the nonlinear solver p (a) density (b) Pressure Fig Eample 5.8 at T =.3, with N = and CFL =. With the positivity-preserving limiter on. Solid line is the eact solution and blue squares are the numerical approimations Concluding remarks. In this paper, we develop an implicit positivity-preserving DG method with high-order spacial accuracy for one-dimensional conservation laws. This work is an etension of the positivity-preserving limiter in [41, 4] for eplicit schemes to implicit ones with backward Euler time discretization. To make the scheme positive, a lower bound for the CFL number is necessary. This conclusion is verified via both theoretical analysis and numerical eperiments. The positivitypreserving limiter not only makes the numerical approimation physically meaningful but also brings robustness to the scheme and accelerates convergence towards the steady-state solution. The scheme also sees its success on the compressible Euler system. In the future, we have the following three directions to further eplore. First, even though the result in this paper easily generalizes to multidimensional tensor product meshes and polynomial spaces, positivity-preserving DG schemes for multidimensional domain with unstructured mesh needs to be further developed. Second, we plan to generalize the proposed implicit positivity-preserving DG scheme to other types of equations such as the convection-diffusion equations. Thirdly, high-order implicit temporal discretizations need to be considered. Since there are no implicit SSP methods with order greater than one [15], we need to turn to other types of implicit time discretizations, such as BDF methods and fully implicit RK methods. Appendi A. A proof for Lemma 3.8. Proof. The first statement is a direct conclusion of the third property of the discrete delta function in Lemma 3.6. For the second statement, let us first derive an eplicit formula for βi k 56. By the 57 Rodrigues formula in Lemma 3.5 we have d l p l () = 1 l l! d l ( 1) l = 1 [ ( 1) l l l! d l ( + 1) l] d l