CFL Conditions for Runge-Kutta Discontinuous Galerkin Methods on Triangular Grids

Transcription

1 CFL Conditions for Runge-Kutta Discontinuous Galerkin Methods on Triangular Grids T. Toulorge a,, W. Desmet a a K.U. Leuven, Det. of Mechanical Engineering, Celestijnenlaan 3, B-31 Heverlee, Belgium Abstract We study time ste restrictions due to linear stability constraints of Runge-Kutta Discontinuous Galerkin methods on triangular grids. The scalar advection equation is discretized in sace by the Discontinuous Galerkin method with either the Lax-Friedrichs flux or the uwind flux, and integrated in time with various Runge-Kutta schemes designed for linear wave roagation roblems or non-linear alications. Von-Neumann-like analyses are erformed on structured eriodic grids made u of congruent elements, to investigate the influence of element shae on the stability restrictions. We assess CFL conditions based on different element size measures, among which only the radius of the inscribed circle and the shortest height rove aroriate, although they are not totally indeendent of the triangle shae. We exlain their general behaviour with resect to element quality, and reort the corresonding Courant numbers with both tyes of flux and olynomial order ranging from 1 to 1, for use as guidelines in ractical simulations. We also comare the erformance of the Lax-Friedrichs flux and the uwind flux, and we draw general conclusions about the relative comutational efficiency of RK schemes. The alication of CFL conditions to two examles involving resectively an unstructured and a hybrid grid confirms our results, although it shows that local stability criteria tend to yield too restrictive conditions. Keywords: Discontinuous Galerkin, Runge-Kutta, Linear Stability, CFL Condition Corresonding author addresses: thomas.toulorge@mech.kuleuven.be (T. Toulorge), wim.desmet@mech.kuleuven.be (W. Desmet) Prerint submitted to Journal of Comutational Physics December 7, 212

2 1. Introduction 1.1. Context Among the numerous numerical methods used to solve hyerbolic artial differential equations on unstructured grids, the Discontinuous Galerkin (DG) Method is receiving growing attention in different fields like comutational electromagnetics, comutational fluid dynamics or comutational aeroecoustics. Its ability to obtain solutions with arbitrarily high order of accuracy is a articularly interesting feature. Other advantages over concurrent high-order methods like Finite Differences are the straightforward formulation of boundary conditions, as well as the comactness of the scheme, that allows an efficient arallel imlementation. When solving time-deendent Partial Differential Equations (PDE s), DG schemes can be combined with Runge-Kutta (RK) time integrators in a method of lines, as introduced by Cockburn and Shu [1]. This aroach, taking advantage of the comutational efficiency of exlicit integration methods for moderately stiff PDE s, has been successfully alied to a broad range of wave roagation and convection-dominated roblems [2, ch. 1,. 11]. As any exlicit method of line, the RKDG method features stability restrictions that are illustrated by the the well-known Courant-Friedrichs-Levy (CFL) inequality: a t h C (1) where a is the magnitude of the largest characteristic quantity of the hyerbolic system, t is the time ste, h is the element size and C is a constant that deends on the satial and time discretization methods. The left-hand side of Inequality (1) is called the Courant number. In ractice, this relation imoses a suerior bound on the time ste, thus limiting the comutational efficiency of the method. With one satial dimension, a DG sace discretization using olynomials of degree, associated to a ( + 1)-stage RK time integrator of order + 1, was formally roven to be stable under Condition (1) with: C = u to = 2 [3], this condition being otimal for = [1] and = 1 [4]. Moreover, numerical evidence was given that these values are less than 5% smaller than the otimal CFL limit for 2 [1]. Kubatko et al. [5] studied the linear stability of stage-exceeding-order SSP RK schemes with DG satial discretization, and gave values for the maximum Courant number, for 3. In the case of multile satial dimensions, the conditional stability of the method was demonstrated in Ref. 6. However, no clear link between the element geometry and the stability bound was ut in evidence. In ractice, engineers and researchers use Condition (1) in 2D and 3D to determine the maximum time ste to be set in their simulations, choosing emirically a measure for the element size h. The analysis of Ref. 5 was extended to 2D [7] for two structured triangular grid configurations, and a grid arameter h was roosed. However, that work does not address the influence of element shae on the stability bounds. In this aer, the relation between the shae of 2D triangular elements and stability restrictions is investigated in a systematic way, with the aim of roviding CFL conditions that could be used to set the time ste in ractical simulations. We describe a method, similar to the one resented in Ref. 7, to analyze numerically the stability of RKDG methods on a grid comosed of congruent elements, so that a given element shae can be associated to a stability limit. This method is alied to a broad range of triangle shaes, and results are resented in the form of values for the maximum Courant number, calculated with different geometrical arameters for the element size h. Conclusions are drawn on the ability of each size measure to take into account the influence of element shae in the CFL condition. The study is reeated for two tyes of inter-element fluxes, namely the Lax-Friedrichs flux and the uwind flux, and several RK schemes that are commonly used with DG sace discretization, in order to rovide fairly general results. 2

3 1.2. Discontinuous Galerkin Method As a model for hyerbolic conservation laws, the scalar advection equation over a domain with eriodic boundary conditions is considered: q t + a rq = (2) x r where q is the unknown, t is the time, x r is the r-th sace coordinate, and a r is the r-th comonent of the constant advection vector a. Einstein s summation convention is used over the r index. For each element Ω resulting from the artitioning of the comutational domain, a basis B Ω = { ϕ Ω j, j = 1...N } is defined, in which the comonents ϕ Ω j are olynomials of order suorted in Ω, with N = (+1)(+2) 2 for triangular elements. An aroximation q Ω of q on Ω is obtained by a rojection on this basis: q Ω (x 1, x 2, t) = N j=1 q Ω j (t) ϕ Ω j (x 1, x 2 ) Alying the Discontinuous Galerkin rocedure to Eq. (2) results in: ˆ N ϕ Ω k Ω j=1 q Ω j t ϕω j dω ˆ Ω ϕ Ω k x r N a r qj Ω ϕ Ω j dω j=1 + ˆ N ϕ Ω k Ω j=1 F Ωi j ϕ Ω j d Ω =, k = 1... N (3) In Eq. (3), F Ωi is an aroximation of the numerical flux comuted on the element edge Ω i that is common to Ω and its neighbour Ω Ωi i. In this work, F is either the Lax-Friedrichs flux: F Ωi LF = 1 [ (a n Ω i ) ( ( q Ω + q i) )] Ω a q Ω i q Ω (4) 2 or the uwind flux: F Ωi Uwind = {( a n Ω i ) q Ω, a n Ωi ( a n Ω i ) q Ω i, a n Ω i < (5) n being the outgoing unit normal to the element edge Ω i. These two choices of flux are the most widely used to solve linear PDE s. Following the quadrature-free form of DGM [8], all elements in the hysical domain can then be maed onto a single reference element : M Ω Ω : (ξ 1, ξ 2 ) (x 1, x 2 ) The Jacobian matrix of M Ω is defined as J Ω ij = xi ξ j, with determinant D Ω. Likewise, each element edge Ω i is maed onto a unique edge i of by a function with Jacobian matrix J Ωi, of determinant D Ω. The basis B Ω can then be defined by alying the transformation M Ω to a unique basis B = {ϕ j, j = 1...N } defined in with reference coordinates (ξ 1, ξ 2 ). The change of variables in Eq. (3) yields: ˆ N ϕ k j=1 qj Ω t ϕ j D Ω d ˆ ( J Ω ) 1 sr + ϕ k ξ s 3 ˆ i=1 N a r qj Ω ϕ j D Ω d j=1 N ϕ k i j=1 3 F Ωi j ϕ j D Ω i d i =, k = 1... N (6)

4 Eq. (6) can be recast in matrix form: M Ω qω t K Ω r a r q Ω + 3 M Ωi F Ωi = (7) i=1 where q Ω and F Ωi are the vectors with comonents qj Ω Ωi and Fj resectively, and: ˆ M Ω kj = ϕ k ϕ j D Ω d ˆ ( ) K Ω r kj = ( J Ω ) 1 ϕ k ϕ sr j D Ω d (8) ξ ˆ s M Ωi kj = ϕ k ϕ j D Ω i d i i 4

5 2. Method 2.1. Stability Analysis Semi-Discrete Discontinuous Galerkin Oerator The global DG sace oerator L can be assembled directly by alying Eq. (7) for all elements of a grid, yielding: q t = L q where q contains the semi-discrete solution for all degrees of freedom in the comutational domain. In this work, stability analyses are erformed with structured grids made u of eriodic atterns of congruent elements, as illustrated in Fig. 1. In this case, a Von Neumann-like rocedure is used as an alternative to the global oerator assembly. It consists in considering harmonic solutions on a single attern: q = ˆq e i(kx x+ky y) and exloiting the eriodicity of atterns to formulate the semi-discrete oerator: ˆq t = L (k x, k y ) ˆq where ˆq reresents the comlex amlitude of the solution for all degrees of freedom in a attern. y x (a) (b) Figure 1: Structured grid (a) and sketch of the eriodic attern of elements (b) used for stability analysis Stability of the Fully Discrete Scheme The stability of a RK method is determined by its characteristic olynomial P, obtained by alying the time integration scheme to the model equation: u t = λu (9) with λ C. The time steing scheme can then be formulated as u n+1 = P (z) u n, where z = λ t, and the absolute stability region S of the RK scheme is given by: S = {z : P (z) 1} 5

6 To evaluate the stability of the fully discrete RKDG method with a given time ste t, the eigenvalues λ k (L) of the semi-discrete oerator L are subsituted to λ in Eq. (9), and the stability condition, illustrated in Fig. 2, becomes: λ k (L) t S, λ k (L) (1) This substitution yields a necessary and sufficient stability condition only if L is normal. In the general case, Relation (1) is only a necessary condition for absolute stability, the sufficient condition being more comlex [9, 1]. However, it rovides an excellent guideline for the choice of t [2, ch. 4,. 95]. In order to find the maximum allowable time ste t max for stability, a simle bisection method iteratively reduces the bracket interval [ t low, t high ] subject to the conditions: { P (λ k t low ) 1, λ k λ k : P (λ k t high ) > RK λ* t 2 1 Im(λ* t) Re(λ* t) Figure 2: Stability lot for the 2D DG sace oerator ( = 1). The circles mark the boundary P (z) = 1 of S Exloration of Triangle Shae To determine the deendence of the stability bound on the triangle shae, the stability analysis rocedure described in Sec. 2.1 is alied to various grids made u of eriodic atterns (see Fig. 1). In each of these grids, all elements are congruent, so that a stability limit can be associated to each element shae. To exlore triangle shaes in a systematic way, consider first that a triangle is uniquely determined by secifying the length of its three sides l 1, l 2 and l 3. Now, consider two similar triangles differing only by a scale factor α: it can be deduced from Eqs. (7) and (8) that the semi-discrete oerator L is inversely roortional to α. Thus, we fix l 1 : l 1 = x = 2 and only two indeendent arameters (l 2, l 3 ) need to be studied. Moreover, l 2 and l 3 are interchangeable, so that only half of the two-arameter sace needs to be exlored: Moreover, the triangle inequalities: l 2 l 3 l 1 l 2 + l 3 l 2 l 1 + l 3 6

7 reduce the region to be exlored as illustrated in Fig 3. To characterize the triangle shaes in a simle manner, we choose a measure γ, that is commonly used for grid quality assessment in meshing methods: γ = 2 r inner r circum where r inner is the radius of the inscribed circle and r circum is the radius of the circumcircle of the triangle. Fig. 4 shows the value of the grid quality measure as a function of l 2 and l 3. A set of 52 oints, indicated in Fig. 4, is chosen in the arameter sace (l 2, l 3 ), sanning a large variety of triangle shaes, as the grid quality measure γ varies from.31 to 1, with a mean of.45. The actual shae of these 52 triangles is illustrated in Fig. 5. l 3 l 2 l 3 l 2 l 1 l 3 l 1 l 2 l 3 l 2 Figure 3: Parameter sace (l 2, l 3 ). The region to be effectively exlored is colored in grey. 1.8 γ l 3 l 2 Figure 4: Mesh quality measure γ in function of the triangle side lengths l 2 and l 3. Black stars reresent the osition of the chosen triangles in the arameter sace (l 2, l 3 ). 7

8 2.3. RKDG Method As shown in Sec , the maximum time ste allowed for stability deends both on the eigenvalues of the semi-discrete oerator λ (L) and the choice of the RK scheme. In this section, we shall exlain which asects of the sace and time discretization methods affect the stability bound, and the choices made to obtain results that can be alied to ractical simulations. Two areas of alication, where the use of RKDG methods is becoming oular, are articularly targeted, namely linear wave roagation roblems and non-linear roblems. It is to be noted that with non-linear equations, conditions for linear stability are usually more restrictive than those for non-linear stability, but the method must be linearly stable to revent round-off errors from ruining the high-order accuracy [1]. Thus, the CFL conditions resented in Sec. 3 may be of interest for both linear and non-linear alications Sace Discretization We can deduce from Eq. (3) that, aart from the grid, the oerator L deends on the basis functions φ j and the numerical flux F Ωi. Nevertheless, it can be shown [11] that the choice of φ j does not influence the eigenvalues λ (L), as long as the basis B Ω sans the same olynomial sace. This means that the stability bound deends only on the order of the olynomial basis. In Sec. 3, results are resented for order from 1 to 1, as most ractical simulations are erformed within this range. The choice of the numerical flux F Ωi strongly influences the stability of the method. The simlest versions are the central flux, the Lax-Friedrichs flux and the uwind flux. However, the central flux yields a conservative scheme (all eigenvalues λ (L) are imaginary) which fails to dam the surious modes of the system, so that it is rarely used in ractical simulations. On the other hand, Lax-Friedrichs flux and the uwind flux are the most common choices for the discretization of linear equations, the former because of its simlicity and low comutational cost, the latter because of its otimal dissiation roerties. These two tyes of numerical fluxes, described in Sec. 1.2, are studied in this work. For non-linear equations, oular choices are the Lax-Friedrichs flux again, the Godunov flux [1] and the Engquist-Osher [12] (or Osher-Salomon [13]) flux. The Godunov flux reduces to the uwind flux in the linear case, so does the Engquist-Osher flux in regions where the sign of the characteristic velocity is constant [12, 13] Time Discretization Numerous RK schemes are used in the framework of the method of lines. The schemes used in this work are summarized in Table 1. The choice of a articular scheme generally results from a trade-off between accuracy and stability, thus it deends on the alication targeted. In this work, we shall focus on two tyes of RK schemes. First, we study a grou of RK schemes commonly used with, or secifically designed for linear wave roagation roblems, for which the DG method is increasingly oular. All of them are fourth-order accurate for linear equations. They are all designed with five or more stages, so that the coefficients for higher stages, that are not used to fulfill the order conditions, rovide extra degrees of freedom for otimization. Carenter and Kennedy [14] were among the first to roose a 2N-storage scheme, that they otimized with resect to stability. Mead and Renaut [15], Allamalli et al. [16] also resented schemes with otimal stability region. Hu et al. [17] devised RK schemes otimized with resect to dissiation and disersion, for which Stanescu and Habashi [18] gave a 2N-storage imlementation. A similar methodology was followed by Berland et al. [19]. Finally, Calvo et al. [2], as well as Tselios and Simos [21], otimized their scheme with resect to both stability and accuracy. It is to be noted that the schemes roosed by Berland et al. [19] and Calvo et al. [2] are almost the same, although they were obtained through different methods. Absolute stability regions of all these schemes are shown in Fig 6. They were all otimized with resect to finite-difference semi-discrete oerators, excet those of Mead and Renaut [15] who used seudo-sectral oerators. As high-order finite-difference methods are usually based central schemes, they are non-diffusive, thus the eigenvalue sectra of the corresonding oerators lie on the imaginary axis. This exlains why the stability region of RK schemes otimized for stability are larger along the imaginary axis, but not necessarily along the real axis. As seen from the stability lot in Fig. 2, the extent of the RK stability region along the real axis is more relevant for RKDG stability, but to our knowledge, no RK method has been secifically designed for DG sace oerators yet. 8

9 Then, we study otimal strong-stability-reserving (SSP) RK schemes, that are used with non-linear alications [22, 23]. To our knowledge, non-linear stability of RKDG methods has only been demonstrated with SSP schemes [1]. The otimal three-stage third-order scheme is classical [22]. For non-linear fourthorder accuracy, a minimum of five stages is required, and the otimal scheme we use is obtained through a numerical otimization rocedure [23]. Absolute stability regions of the SSP RK schemes studied in this work are shown in Fig. 7. Finally, simle low-storage RK schemes [24] with + 1 stages are used for comarison urose. They yield a formally ( + 1)-order RKDG method u to fourth order. Their stability regions are lotted in Fig. 8. One can note that the 3-stage and 4-stage schemes are equivalent to the (3,3)-SSP and standard fourth-order RK schemes resectively, as all s-stage, s th -order RK schemes have the same stability region. Name Order Stages Storage Observations Carenter [14] 4 5 2N Otimized for stability with high-order FD in CFD Mead RKC [15] 4 6 5N Otimized for stability at fixed accuracy, with seudosectral methods HALE-RK6 [16] 4 6 2N HALE-RK7 [16] 4 7 2N Otimized for stability with high-order FD HALE-RK67 [16] N Hu LDDRK6 [17] 4 6 3N Otimized for accuracy with high-order Hu LDDRK46 [17] N FD. 2N-storage imlementation of Hu LDDRK56 [17] N LDDRK6 and LDDRK56 available [18] Berland [19] 4 6 2N Otimized for high accuracy Calvo LDDRK46 [2] 4 6 2N Otimized for stability and accuracy with non-dissiative satial discretizations Tselios DDAS47 [21] 4 7 2N Otimized for stability and accuracy for non-dissiative satial discretizations Otimal (3,3)-SSP [22] 3 3 3N Used in RKDG methods for non-linear Otimal (5,4)-SSP [23] 4 5 5N roblems Jameson [24] Variable +1 2N Table 1: Main characteristics of RK schemes used in this work. 9

10 y (a) y (b) Figure 5: Triangle shaes used for the stability analysis: global view (a) and zoom (b). 1

11 Carenter Mead HALE-RK6 HALE-RK7 HALE-RK Berland Hu LDDRK6 Hu LDDRK46 Hu LDDRK (a) (b) 5 4 Calvo LDDRK46 Tselios DDAS (c) Figure 6: Absolute stability regions of RK schemes for linear wave roagation roblems, otimized with resect to stability (a), to accuracy (b), and to both stability and accuracy (c). 11

12 4 3 Otimal SSP (3,3) Otimal SSP (5,4) Figure 7: Absolute stability regions of SSP RK schemes Stage 3-Stage 4-Stage 5-Stage 6-Stage Stage 8-Stage 9-Stage 1-Stage 11-Stage (a) (b) Figure 8: Absolute stability regions of Jameson ( + 1)-stage RK schemes, 2 to 6 stages (a), and 7 to 11 stages (b). 12

13 3. Results 3.1. Advection Velocity and Numerical Flux Before studying the CFL conditions, it is interesting to qualitatively assess the influence of the advection velocity on stability. It can be seen from Eqs. (7), (4) and (5) that L is roortional to the advection velocity a, so that we set a = 1, and study only the effect of the advection direction θ with a = (cos θ, sin θ). Additionally, we can deduce from symmetry considerations that the roblem is invariant with resect to the sign of a (i.e. the results are π-eriodic in θ). In this Section, we resent results obtained with Jameson RK, but the same behavior is found with other RK schemes. Fig. 9 shows the mesh attern and maximum time ste t max in function of the advection direction θ for an equilateral triangle (which is the triangle shae of highest quality, γ = 1). One can see that the time ste is almost constant for the Lax-Friedrichs flux, whereas the deendence on θ is stronger but moderate for the uwind flux. The same lots are resented in Fig. 1 for a triangle of lower quality. The time ste variation in function of θ remains moderate with the Lax-Friedrichs flux, but with uwind flux the time ste becomes much larger in the advection direction arallel to the longest triangle sides. We verified that this behavior is qualitatively the same for higher order. In ractical simulations, the governing system of multidimensional hyerbolic equations often has characteristics that do not degenerate into lines. Instead, there is an infinite set of characteristic directions forming a Monge cone. Thus, a unique advection direction cannot be identified for each characteristic variable, and all advection directions have to be considered. This is the case, for instance, with the acoustic modes featured by the Euler equations in Fluid Dynamics. For the assessment of the CFL conditions in Sec. 3.2, we thus consider only the minimum value t max of the time ste with resect to θ: t max = min θ ( t max ) which is the value that ensures stability for all advection directions. For this urose, we let θ swee the range [, 18 ] with a ste of 4, and a stability analysis is carried out for each value of a. By comaring Fig. 9 and 1, one can observe that the time ste t max is significantly greater with the uwind flux than with the Lax-Friedrichs flux for all advection directions in the case of the equilateral triangle, whereas the minimum value, t max, seems to be almost equal with both tyes of numerical fluxes in the case of the stretched triangle. We verified that the uwind flux yields a greater or equal time ste in all cases. The maximum and minimum relative difference in t max of the set of triangles studied are defined as: ( ) t max,uwind t max,lf max = max γ t max,lf ( ) t max,uwind t max,lf min = min γ t max,lf where max γ and min γ denote resectively the maximum and minimum of a quantity with resect to the element shae, that is, the maximum and minimum value in the set of triangles studied. The quantities max and min are lotted for all RK schemes and order from 1 to 1 in Fig. 11. The maximum difference max, that varies between 25% and 6%, is always obtained with the equilateral triangle (i.e. the best quality element). The minimum difference min, increasing from % at = 1 to 33% at = 1 for most RK schemes, is given by high-asect-ratio triangles. The otimal (4,5)-SSP RK, and to a lesser extent the Carenter and Hu LDDRK46 RK, seem to give less advantage to the uwind flux than the other RK schemes studied in this work. They yield a articularly small minimum difference in t max at low order, which occur on highly stretched triangles. This is imortant in view of ractical simulations, when the uwind flux requires additional comutational effort er time ste, because of the characteristic decomosition of the governing system of equations that it involves. Then the uwind flux may not be cometitive with the Lax-Friedrichs flux in terms of comutational efficiency, esecially at low order with low-quality grids, when using these less advantageous RK schemes. 13

14 Lax-Friedrichs Uwind Lax-Friedrichs Uwind t max (a) θ (b) (c) Figure 9: Time ste in function of the advection direction θ with Jameson RK for an equilateral triangle at = 1: olar (a) and cartesian (b) lots, corresonding mesh attern (c). 14

15 Lax-Friedrichs Uwind 7 6 Lax-Friedrichs Uwind t max (a) θ (b) (c) Figure 1: Time ste in function of the advection direction θ with Jameson RK for a triangle of higher asect ratio at = 1: olar (a) and cartesian (b) lots, corresonding mesh attern (c). max (%) Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson min (%) Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson (a) (b) Figure 11: Relative maximum (a) and minimum (b) difference in t max between uwind and Lax-Friedrichs fluxes in the set of triangles studied. 15

16 3.2. CFL Conditions From t max, the maximum Courant number ν can now be calculated: ν (a) = a t max h and we consider the minimum value ν over all advection directions, as exlained in Sec. 3.1: ν = a t max h In this work, we study five different element size measures h: the shortest edge in the triangle (ν l ), the side length of the equilateral triangle with equal area (ν e ), the radius of the circumscribed circle (ν c ), the radius of the inscribed circle (ν r ) and the shortest height in the triangle (ν h ). Courant numbers based on the radius of the inscribed circle are very commonly used in Finite Volume and Finite Element Methods. Obviously, the ideal CFL condition would yield the same maximum Courant number whatever the shae of the element. However, none of the five element size measures manages to reflect erfectly the influence of element shae on the stability bound, and there is disersion among the 52 values of ν, corresonding to the 52 triangles studied, for a given size measure. We denote by min γ (ν ) and max γ (ν ) resectively, the minimum and maximum of ν with resect to the element shae, that is, the minimum and maximum values of ν in the set of triangles studied. The value min γ (ν ), which ensures stability for any element shae, is the safest choice for use in ractical simulations, but one may want to use higher values in secial cases, as exlained in Sec In Aendix A, we rovide values of min γ (ν ) and max γ (ν ) for ν based on the most interesting element size measures, with the two kind of numerical fluxes and order ranging from 1 to 1, for all the RK schemes described in Sec We also give the values of the coefficients α, α 1 and α 2 for the curve: ln ( ) = α + α 1 ln () + α 2 [ln ()] 2 (11) that is found to fit the data for min γ (ν ) and max γ (ν ) as a function of, with a maximum error of aroximately 1%. To characterize the deendence of the CFL conditions on the element shae, we comute the deviation of ν with resect to the minimum value in the set of triangles studied: D = max γ (ν ) min γ (ν ) min γ (ν ) Element Size Measures Fig. 12 shows the value of ν based on each of the five element size measures, for all triangles, at order = 1, with the Jameson RK scheme. One can see that νl, ν e and νc, based resectively on the shortest side length, the side length of the equilateral triangle with equal area and the radius of the circumscribed circle, exhibit large relative deviations with resect to their minimum value. This is due to the fact that these measures do not take low values for flat, high-asect-ratio triangles (under the condition that the three side length are have the same order of magnitude for νl ), whereas such ill-conditioned elements yield small t max. As they fail to correctly characterize such athological cases, the results obtained with the set of triangles described in Sec. 2.2 cannot be generalized, and they are not aroriate for use with arbitrary unstructured grids. Obviously, the argument stated above being based on geometry, the same behaviour is observed at higher olynomial order and with other RK schemes. On the contrary, νr and νh exhibit less relative variation with the element shae, for both tyes of numerical fluxes. Therefore, only the results for νr and νh are resented in Aendix A. As seen in Fig. 12, νr increases with γ (more for the uwind flux than for the Lax-Friedrichs flux), whereas νh decreases (Lax-Friedrichs flux) or remains almost constant (uwind flux). We verified that the same behaviour is obtained for all RK schemes and all orders studied in this work. These remarks can be exloited to finetune the value of ν set in ractical simulations. In some alications, the global time ste is more likely to be limited by small or medium-size low-quality elements (for instance the elements used to mesh the boundary 16

17 layer in CFD) than by small, high-quality elements: then minimum values of νr are otimal, whereas with the Lax-Friedrichs flux, higher values of νh, close to max γ (νh ), are aroriate. If, on the other side, a grid has uniformly good quality, then minimum values of νr are sub-otimal, whereas minima of νh suit the Lax-Friedrichs flux. Fig. 13 shows the deviation D in νr and νh with the Lax-Friedrichs flux. D varies between 7% and 31% for νr, deending on the order and the RK scheme used, whereas the deviation is generally higher (between 2% and 42%) for νh. However, one can see that the CFL condition based on the inner radius is more accurate only u to order = 6, the shortest height becoming a generally better size measure for higher order. The same quantities are lotted in Fig. 14 for the uwind flux. Surrisingly, the commonly-used inner radius yields a relatively inaccurate CFL condition (52% to 63% deviation), whereas the shortest height measure is very reliable (4% to 13% deviation) ν l ν e ν c νr ν h.8.7 ν l ν e ν c νr ν h ν.3.25 ν γ γ (a) (b) Figure 12: Minimum value ν over all advection directions of the maximum Courant number ν, in function of the mesh quality measure γ, at order = 1, with the Lax-Friedrichs flux (a) and the uwind flux (b), for Jameson RK scheme. D (%) Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland 15 Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 1 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson D (%) Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson (a) (b) Figure 13: Deviation D, among all element shaes studied, of ν based on the inner radius (a) and the shortest height (b), in function of the order, with the Lax-Friedrichs flux, for all RK schemes. 17

18 Runge-Kutta Schemes As seen in Fig. 13 and 14, the CFL conditions have similar accuracy with most RK schemes, for a given tye of numerical flux, a given element size measure and a given order. Nevertheless, one can note that the shortest height measure is more aroriate for the otimal (4,5)-SSP RK scheme whatever the order, for the Lax-Friedrichs flux. Also, the CFL conditions seem to be generally slightly less accurate with the otimal (4,5)-SSP and the Carenter RK schemes than with other RK schemes for the uwind flux. A general comarison the comutational efficiency based on min γ (ν ) is difficult, as this minimum value is obtained with different triangle shaes deending on the numerical flux, the element size measure and sometimes even on the order. Thus, the exact answer to the question of which RK scheme minimizes the comutation time for a ractical simulation is grid-secific. However, trends can be observed in the results reorted in Aendix A. As the comutational effort of a RK scheme in one time ste is roortional to its number of stages, the Courant number er stage is taken a measure of the efficiency. Fig. 15 and 16 show the minimum value of the maximum Courant numbers νr and νh er stage with the Lax-Friedrichs flux and the uwind flux resectively. In all cases, it can be seen that the Carenter RK and the otimal SSP RK schemes are most efficient, which seems logical as they are all otimized for stability. The Hu LDDRK, Berland RK, Calvo RK and Tselios RK schemes, designed (at least artially) for high accuracy, are comutationally less efficient. The resence of the HALE RK and Mead RKC schemes among the least efficients, although they are otimized for stability, demonstrates that otimizing a RK scheme with resect to Finite Difference or Pseudo-Sectral satial oerators does not necessarily yield the exected results with the DG method, as exlained in Sec Finally, the Jameson scheme is relatively efficient for order 2, but it becomes less interesting for higher order. In articular, for > 3, its number of stages increases without necessarily roviding higher order, and the free coefficients are not otimized for anything. The values of min γ (νr ) given in Tables A.6 and A.12, as well as Fig. 15 and 16, are barely higher with the uwind flux than with the Lax-Friedrichs flux for moderate order. This is because the minima of ν based on the inner radius are obtained on bad-quality elements, where the uwind flux is less advantageous, as exlained in Sec

19 D (%) Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson D (%) Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson (a) (b) Figure 14: Deviation D, among all element shaes studied, of ν based on the inner radius (a) and the shortest height (b), in function of the order, with the uwind flux, for all RK schemes. 19

20 min γ ν er stage Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson min γ ν er stage Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson (a) (b) min γ ν er stage Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson min γ ν er stage Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson (c) (d) Figure 15: Minimum value min γ (ν ) of ν er stage, based on the inner radius (a) and (b), and based on the shortest height (c) and (d), in function of the order, with the Lax-Friedrichs flux, for all RK schemes. 2

21 min γ ν er stage Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson min γ ν er stage Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson (a) (b) min γ ν er stage Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson min γ ν er stage Carenter Hu LDDRK6 Hu LDDRK46 Hu LDDRK56 Berland Calvo LDDRK46 Tselios DDAS47 HALE-RK6 HALE-RK7 HALE-RK67 Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson (c) (d) Figure 16: Minimum value min γ (ν ) of ν er stage based on the inner radius (a) and (b), and based on the shortest height (c) and (d), in function of the order, with the uwind flux, for all RK schemes. 21

22 4. Examles In order to illustrate the alication of the results obtained in Sec. 3, we comute the maximum time ste t max allowed for stability with two different triangular grids, that are more reresentative of ractical roblems than the structured grids described in Sec The exact t max is obtained by considering eriodic boundary conditions and assembling directly the semi-discrete oerator L for the whole grid, as exlained in Sec The minimum value over all advection directions is then comared to the CFL conditions based on the inner radius and on the shortest height. As the advection velocity is uniform over the whole comutational domain, the most restrictive value of t max comuted by the CFL conditions is obtained for the smallest element in the grid. We show results obtained with both minimum values min γ ν (Tables A.6, A.9, A.12 and A.15) and maximum values max γ ν (Tables A.7, A.1, A.13 and A.16) of ν given in Aendix A. The first grid, shown in Fig. 17, is unstructured and contains 334 triangular elements. Its quality can be considered as uniformly good, with γ ranging from.78 to 1. The results of time ste calculations at order = 4 are given in Tables 2 for the Lax-Friedrichs flux and Table 3 for the uwind flux. With the Lax-Friedrichs flux, the CFL condition based on the inner radius rovides better time stes for high values of the Courant number νr than for lower values, as exected from the discussion on good-quality grids in Sec On the other side, the time stes obtained by using maximum values of νh for the CFL condition based on the shortest height exceed the stability limit with most of the RK schemes, as then νh is globally decreasing with γ. With the uwind flux, both elements size measures give better results for high values of the Courant number ν, in accordance with the observations in Sec The second grid, shown in Fig. 18, is made u of a structured art and an unstructured art, like those commonly used to resolve boundary layers in CFD or CAA alications. It contains 164 triangles of heterogeneous quality (.13 < γ < 1), the worst elements being located in the structured art. The results of time ste calculations at order = 5 are given in Tables 4 and 5. For the Lax-Friedrichs flux, the minimum value of νr and maximum value of νh yield accurate results, as they are suited to the highasect-ratio elements of the structured zone that restrict the time ste on this grid. The minimum value of νh is erforms worse, and the maximum value of ν r leads to a violation of the exact stability restriction, in accordance with the conclusions of Sec For the uwind flux, only the maximum value of νr yields reasonably good accuracy, whereas it should give overestimated time stes, according to the discussion in Sec In these two examles, the relative accuracy of different CFL conditions behaves in accordance with the qualitative observations of Sec with resect to the tye of flux, the kind of grid, and the range of Courant number used. However, the general level of accuracy can considered as disaointing in view of the results of Sec (time stes of less than half of the otimal time ste with the hybrid grid and the uwind flux, for instance). The main reason for this lack of accuracy is not a misrediction in the influence of the element shae on the time ste restriction. It is due to the fact that local criteria, such as the CFL conditions, can only rovide bounds for stability, and the global stability condition may be less restrictive. This is even more the case when other tyes of boundary conditions than eriodicity are imosed, because they generally add constraints to the solution, which may eliminate some of the unstable modes. Nevertheless, the accuracy of CFL conditions is found to be mainly indeendent of the RK scheme used, and the uwind flux yields greater time stes than the Lax-Friedrichs flux, as in Sec

23 Figure 17: Unstructured grid. ν r Exact Min. Max. Min. Max. Carenter (31.4%) 5.56 (18.2%) 5.21 (23.3%) 6.81 (.3%) Hu LDDRK (31.4%) 4.15 (18.%) 3.9 (22.9%) 5.9 (.5%) Hu LDDRK (31.3%) 3.84 (18.%) 3.61 (22.9%) 4.71 (.6%) Hu LDDRK (31.5%) 4.6 (18.3%) 3.83 (22.9%) 5. (.6%) Berland (31.4%) 4.86 (18.2%) 4.58 (23.%) 5.99 (.7%) Calvo LDDRK (31.3%) 4.9 (18.1%) 4.62 (22.9%) 6.4 (.9%) Tselios DDAS (31.3%) 6. (18.1%) 5.63 (23.2%) 7.38 (.8%) HALE-RK (31.3%) 3.86 (18.2%) 3.63 (22.9%) 4.74 (.5%) HALE-RK (31.4%) 4.74 (18.1%) 4.45 (22.9%) 5.83 (.9%) HALE-RK (31.6%) 4.8 (17.9%) 3.83 (22.9%) 5. (.6%) Mead RKC (31.2%) 4.24 (18.2%) 4. (22.9%) 5.21 (.5%) Ot. (3,3)-SSP (31.2%) 3. (18.2%) 2.83 (22.9%) 3.69 (.6%) Ot. (4,5)-SSP (38.4%) 6.5 (22.1%) 5.68 (27.%) 7.12 (8.4%) Jameson (31.5%) 3.84 (18.2%) 3.62 (22.9%) 4.72 (.6%) Table 2: Maximum time ste t max 1 for the unstructured grid shown in Fig. 17, at order = 4, with the Lax-Friedrichs flux: exact value, estimations obtained by minimum and maximum values of ν for CFL conditions based on the inner radius (νr ) and on the shortest height (ν h ). The relative error with resect to the exact value is indicated in arenthesis, with a negative number if the exact value is exceeded. ν h 23

24 ν r Exact Min. Max. Min. Max. Carenter (52.3%) 7.77 (24.4%) 6.76 (34.2%) 7.38 (28.2%) Hu LDDRK (51.9%) 6.25 (25.2%) 5.63 (32.7%) 5.88 (29.6%) Hu LDDRK (51.5%) 5.58 (24.9%) 4.98 (32.9%) 5.42 (27.%) Hu LDDRK (51.9%) 6.14 (25.1%) 5.52 (32.7%) 5.78 (29.6%) Berland (51.8%) 7.35 (25.2%) 6.61 (32.7%) 6.92 (29.6%) Calvo LDDRK (51.7%) 7.41 (25.1%) 6.66 (32.7%) 6.97 (29.5%) Tselios DDAS (51.8%) 9.7 (25.1%) 8.16 (32.6%) 8.52 (29.6%) HALE-RK (51.9%) 5.83 (25.1%) 5.21 (33.1%) 5.47 (29.8%) HALE-RK (51.8%) 7.15 (25.2%) 6.4 (33.%) 6.71 (29.8%) HALE-RK (51.9%) 6.14 (25.2%) 5.52 (32.7%) 5.78 (29.6%) Mead RKC (51.8%) 6.42 (25.1%) 5.78 (32.5%) 6.4 (29.5%) Ot. (3,3)-SSP (51.6%) 4.54 (25.%) 4.7 (32.8%) 4.27 (29.4%) Ot. (4,5)-SSP (52.4%) 7.63 (24.2%) 6.61 (34.3%) 7.23 (28.2%) Jameson (51.7%) 5.82 (25.1%) 5.21 (32.8%) 5.47 (29.5%) Table 3: Maximum time ste t max 1 for the unstructured grid shown in Fig. 17, at order = 4, with the uwind flux: exact value, estimations obtained by minimum and maximum values of ν for CFL conditions based on the inner radius (νr ) and on the shortest height (νh ). The relative error with resect to the exact value is indicated in arenthesis, with a negative number if the exact value is exceeded. ν h Figure 18: Hybrid grid. 24

25 ν r Exact Min. Max. Min. Max. Carenter (9.7%) 1.28 ( 11.9%).883 (22.8%) 1.12 (2.4%) Hu LDDRK (9.7%).956 ( 12.3%).657 (22.8%).831 (2.4%) Hu LDDRK (11.2%).884 ( 1.%).67 (24.5%).769 (4.3%) Hu LDDRK (9.9%).934 ( 11.8%).645 (22.8%).815 (2.5%) Berland (9.9%) 1.12 ( 12.1%).772 (22.8%).975 (2.5%) Calvo LDDRK (9.9%) 1.13 ( 11.9%).777 (22.8%).982 (2.5%) Tselios DDAS (9.9%) 1.38 ( 11.9%).951 (22.8%) 1.2 (2.3%) HALE-RK (9.9%).89 ( 12.3%).612 (22.8%).773 (2.5%) HALE-RK (9.6%) 1.9 ( 11.9%).75 (22.8%).949 (2.4%) HALE-RK (9.9%).939 ( 12.4%).646 (22.7%).815 (2.5%) Mead RKC (9.9%).978 ( 12.1%).673 (22.8%).85 (2.5%) Ot. (3,3)-SSP (1.1%).692 ( 12.2%).477 (22.7%).61 (2.5%) Ot. (4,5)-SSP (16.1%) 1.38 ( 5.3%).948 (27.6%) 1.19 (9.%) Jameson (1.%).978 ( 12.1%).674 (22.7%).851 (2.5%) Table 4: Maximum time ste t max 1 for the hybrid grid shown in Fig. 18, at order = 5, with the Lax-Friedrichs flux: exact value, estimations obtained by minimum and maximum values of ν for CFL conditions based on the inner radius (νr ) and on the shortest height (νh ). The relative error with resect to the exact value is indicated in arenthesis, with a negative number if the exact value is exceeded. ν h ν r Exact Min. Max. Min. Max. Carenter (57.2%) 1.78 (32.%) 1.15 (56.2%) 1.24 (52.7%) Hu LDDRK (55.%) 1.44 (3.%).946 (54.1%) 1.1 (51.1%) Hu LDDRK (56.1%) 1.28 (3.4%).825 (55.1%).892 (51.5%) Hu LDDRK (55.1%) 1.42 (29.9%).926 (54.2%).991 (51.%) Berland (55.1%) 1.69 (3.1%) 1.11 (54.1%) 1.18 (51.3%) Calvo LDDRK (54.9%) 1.71 (29.9%) 1.12 (54.2%) 1.19 (51.1%) Tselios DDAS (55.%) 2.9 (29.9%) 1.36 (54.3%) 1.46 (5.8%) HALE-RK (55.1%) 1.35 (3.%).882 (54.1%).941 (51.%) HALE-RK (55.1%) 1.65 (29.9%) 1.8 (54.2%) 1.16 (5.7%) HALE-RK (55.2%) 1.42 (29.9%).927 (54.1%).992 (5.9%) Mead RKC (55.%) 1.48 (3.1%).97 (54.1%) 1.3 (51.%) Ot. (3,3)-SSP (55.5%) 1.4 (29.6%).672 (54.7%).727 (5.9%) Ot. (4,5)-SSP (57.4%) 1.75 (32.2%) 1.12 (56.7%) 1.2 (53.3%) Jameson (55.1%) 1.48 (3.1%).97 (54.1%) 1.3 (51.1%) Table 5: Maximum time ste t max 1 for the hybrid grid shown in Fig. 18, at order = 5, with the uwind flux: exact value, estimations obtained by minimum and maximum values of ν for CFL conditions based on the inner radius (νr ) and on the shortest height (νh ). The relative error with resect to the exact value is indicated in arenthesis, with a negative number if the exact value is exceeded. ν h 25

26 5. Conclusion In this work, we studied the time ste restrictions that arise from RKDG discretizations of the scalar advection equation on triangular grids. Two kinds of numerical fluxes, namely the Lax-Friedrichs flux and the uwind flux, as well as a set of RK schemes targetting both linear wave roagation and non-linear alications, were considered. We alied Von-Neumann-like analysis to structured eriodic grids to derive the linear stability conditions that restrict the time ste. These grids, made u of congruent elements, are articularly aroriate to investigate the influence of triangle shae on the time ste limitations. We focused on the most restrictive condition over all advection directions, which is relevant for generalizing the results to the systems of hyerbolic equations used in ractical simulations. We confirmed that the uwind flux generally allows larger time stes than the Lax-Friedrichs flux. CFL conditions, based on various element size measures, were derived. Only two of them, based resectively on the inner radius and on the shortest height, were found to be aroriate for time ste calculation in ractical simulations, although they are not totally indeendant of the element shae. The corresonding values of the Courant number, as well as their general behaviour with resect to the element shae, are reorted. We verified that this general behaviour is not strongly affected by the choice of the RK scheme. A general icture of the relative merits of the RK schemes in terms of comutational efficiency was deduced from the results. The alication of these results to two examles, involving resectively an unstructured grid and a hybrid grid, confirmed these conclusions. However, it showed that the global stability condition may be much less restrictive than the one given by local criteria such as the CFL conditions. Most of RK schemes studied in this work were otimized for use with finite difference or seudo-sectral sace discretization methods. One can wonder whether a significant gain in comutational efficiency could be obtained with a RK scheme secially designed for DG satial oerators, which has not been done yet, to our knowledge. It is also imortant to note that the asects of comutational efficiency that are addressed in this work are solely related to the stability restrictions. They do not deal with accuracy, which is another imortant criterion for the choice of a time integration method, articularly when using high-order sace discretizations. 26

27 ACKNOWLEDGEMENTS The authors are thankful to Prof. Jean-François Remacle (Université Catholique de Louvain, Belgium) and to Prof. Johan Meyers (K.U. Leuven, Belgium) for valuable discussions about the stability of RKDG methods. The useful suggestions of Drs. Wim De Roeck and Maarten Hornikx (K.U. Leuven, Belgium) are areciated. The anonymous reviewers are gratefully acknowledged for their constructive inut. The work of Thomas Toulorge has been financially suorted by the Euroean Commission through the Marie-Curie Research and Training Network AETHER, Contract Nr. MRTN-CT

28 Aendix A. Maximum Courant Number Results Aendix A.1. Results for the Lax-Friedrichs Flux Carenter Hu LDDRK Hu LDDRK Hu LDDRK Berland Calvo LDDRK Tselios DDAS HALE-RK HALE-RK HALE-RK Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson Table A.6: Minimum value min γ (ν r ) of ν r based on the inner radius, with the Lax-Friedrichs flux Carenter Hu LDDRK Hu LDDRK Hu LDDRK Berland Calvo LDDRK Tselios DDAS HALE-RK HALE-RK HALE-RK Mead RKC Ot. (3,3)-SSP Ot. (4,5)-SSP Jameson Table A.7: Maximum value max γ (ν r ) of ν r based on the inner radius, with the Lax-Friedrichs flux. 28