Stable and compact finite difference schemes

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1 Center for Turbulence Research Annual Research Briefs Stable an compact finite ifference schemes By K. Mattsson, M. Svär AND M. Shoeybi. Motivation an objectives Compact secon erivatives have long been known to have goo accuracy properties for pure secon erivatives. However, for many equations subject to bounary conitions, stability can not easily be proven for problems with a combination of mixe ( 2 / x y) an pure ( 2 / x 2, 2 / y 2 ) secon erivatives, such as the compressible Navier-Stokes equations. We remark that spatial Paé iscretizations (see, for example, Lele (992)) are often referre to as compact schemes. The approximation of the erivative is obtaine by solving a tri- or penta-iagonal system of linear equations at every time step. Hence, Paé iscretizations lea to full ifference stencils, similar to spectral iscretizations. In this paper the term compact will be use exclusively for schemes with a minimal stencil with. For the continuous problem one can erive an energy estimate for the linearize an symmetrize Navier-Stokes equations, proving bouneness of the initial-bounary value problem (see for example Nörstrom & Svär (2005) an Carpenter et al. (999)). Although the analysis is one for 2-D problems, the extension to 3-D problems is straightforwar. If first-erivative ifference operators that satisfy a Summation-By-Parts (SBP) formula (see Kreiss & Scherer (974)) are employe twice for all secon-erivatives (pure an mixe), yieling a non-compact stencil, an if the Simultaneous Approximation Term (SAT) metho by Carpenter et al. (994) is use to implement the bounary conitions, one can exactly mimic the continuous energy estimate (proving stability). There are two obvious rawbacks to this approach when compare to a compact formulation, namely: () There is no mechanism to amp the highest frequency moe (spurious oscillations). (2) Less accurate ifference approximations of pure secon-erivative terms (ue to the leaing orer error constant) are create. The former coul partially be resolve by the aition of artificial amping, but it is ifficult to tell a priori how much is neee. Accuracy an stability are closely linke. The two rawbacks above work together to make compact schemes more accurate than non-compact schemes, especially in regions where viscous effects are important. A property that is not aresse in this paper is computational cost. For a scalar problem on a Cartesian gri, the compact scheme is clearly less expensive, but for more realistic applications (for example the 3-D Navier-Stokes equations on a curvilinear gri), this is still an open question. Compact secon-erivative SBP operators that are stable in combination with firsterivative SBP approximations were erive in Mattsson & Norström (2004). However, it is not clear how to obtain a stable an accurate overall scheme with the aition of mixe erivative terms. In Section 2 we iscuss the SBP property for the first- an secon-erivative ifference operators. A 2-D moel for the Navier-Stokes equations is introuce in Section 3, an we show how to combine the SAT metho an the SBP operators to obtain stable finite

2 22 K. Mattsson, M. Svär & M. Shoeybi ifference approximations using the energy metho (see for example Gustafssonet al. (995)). In Section 4 the accuracy of the compact an the non-compact formulations are compare by performing numerical simulations for the both the moel-problem an the 2-D Navier-Stokes equations. Conclusions are presente in Section Definitions The two-imensional schemes are constructe using -D SBP finite-ifference operators. We begin with a short escription an some efinitions (for more etails, see Kreiss & Scherer (974); Stran (994); Mattsson & Norström (2004)). 2.. One-imensional problems Let the inner prouct for real-value functions u, v L 2 [0, ] be efine by (u, v) = 0 u v x, an let the corresponing norm be u 2 = (u, u). The omain (0 x ) is iscretize using N+ equiistant gri points, x i = i h, i = 0,..., N, h = N. The approximative solution at gri point x i is enote v i, an the iscrete solution vector is v T = [v 0, v,, v N ]. Similarly, we efine an inner prouct for iscrete realvalue vector functions u, v R N+ by (u, v) H = u T H v, where H = H T > 0, with the corresponing norm v 2 H = vt H v. The following vectors will be use frequently: e 0 = [, 0,..., 0] T, e N = [0,..., 0, ] T. (2.) Consier the hyperbolic scalar equation u t + u x = 0 (excluing the bounary conition). Multiplying by u an integration by parts (referre to as the energy metho) leas to t u 2 = (u, u x ) (u x, u) = u 2 0, (2.2) where u 2 0 u2 (x = ) u 2 (x = 0). Definition 2.. A ifference operator D = H Q approximating / x is a compact first-erivative SBP operator if H = H T is iagonal, x T Hx > 0, x 0, an Q + Q T = B = iag (, 0..., 0, ). A semi-iscretization of u t + u x = 0 is v t + D v = 0. Multiplying by v T H from the left an aing the transpose lea to t v 2 H = (v, H Qv) H (H Qv, v) H = v T (Q + Q T )v = v 2 0 v 2 N. (2.3) Equation (2.3) is the iscrete analog of (2.2). For parabolic problems, we nee an SBP operator for the secon erivative. Consier the heat equation, u t = u xx. Multiplying by u an integration by parts leas to t u 2 = (u, u xx ) + (u xx, u) = 2uu x 0 2 u x 2. (2.4) Definition 2.2. A ifference operator D 2 = H ( M + BS) approximating 2 / x 2 is sai to be a compact secon-erivative SBP operator if H = H T is iagonal, x T Hx > 0, x 0, M = M T is sparse, x T Mx 0, S inclues an approximation of the firsterivative operator at the bounary an B = iag (, 0..., 0, ).

3 y NW Compact schemes 23 North NE West East v 0 v v N SW South SE x Figure. Domain 2-D In Mattsson & Norström (2004), high-orer compact secon-erivative SBP operators were constructe. A semi-iscretization of u t = u xx is v t = D 2 v. Multiplying by v T H an aing the transpose, lea to t v 2 H = 2v N (Sv) N 2v 0 (Sv) 0 2v T Mv. (2.5) Formula (2.5) is the iscrete analog of (2.4). Obtaining energy estimates for schemes utilizing both D an D 2 requires that both are base on the same norm H Two-imensional omains We begin by introucing the Kronecker prouct c 0,0 D c 0,q D C D =.., c p,0 D c p,q D where C is a p q matrix an D is an m n matrix. Two useful rules for the Kronecker prouct are (A B)(C D) = (AC) (BD) an (A B) T = A T B T. Next, consier the omain Ω efine as 0 x, 0 y with an (N +) (M +)- point equiistant gri as x i = ih x, i = 0,..., N, h x = N, y j = jh y, j = 0,..., M, h y = M. The numerical approximation at gri point (x i, y j ) is enote v i,j. We efine a iscrete solution vector v T = [v 0, v,..., v N ], where v k = [v k,0, v k,,..., v k,m ] is the solution vector at x k along the y irection, illustrate in Fig.. To simplify the notation we introuce v w, e, s, n to efine the bounary values at the west, east, south an north bounaries (see Fig. ). In orer to istinguish whether a ifference operator P is working in the x or the y irection, we will use the notations P x an P y. The following two-

4 24 K. Mattsson, M. Svär & M. Shoeybi imensional operators will frequently be use: D x = (D I y ), D y = (I x D ), D 2x = (D 2 I y ), D 2y = (I x D 2 ), H x = (H I y ), H y, = (I x H), (2.6) where D, D 2, an H are the one-imensional operators. I x, y are the ientity matrices of appropriate sizes in the x an y irection, respectively. We also introuce the twoimensional norm H H x H y. 3. Numerical metho Our main interest is the compressible Navier-Stokes equations, which can be written as: u t + (Au) x + (Bu) y = C u xx + C 2 u xy + C 2 u yx + C 22 u yy, [x, y] Ω, t 0. (3.) The non-linear equations can be state as (3.) but we consier (3.) to be the linearize, symmetrize, an frozen coefficient equations. It can be shown (see Strang (964)) that if the frozen coefficient problem is well-pose so the non-linear problem will be for smooth solutions. These equations have been stuie more extensively (see for example Nörstrom & Svär (2005)). 3.. The continuous moel problem As a moel of (3.) we consier the two-imensional non-linear parabolic problem u t + ( u2 2 ) x + ( u2 2 ) y = c u xx + c 2 u xy + c 2 u yx + c 22 u yy + F, [x, y] Ω, t 0, (3.2) where F is a forcing function. Equation (3.2) is subject to the following bounary conitions: α w u + c u x + c 2 u y = g w α e u + c u x + c 2 u y = g e α s u + c 2 u x + c 22 u y = g s α n u + c 2 u x + c 22 u y = g n. (3.3) The subscripts enote (w)est, (e)ast, (s)outh an (n)orth bounaries, respectively. The main focus of this paper is to analyze the iscretization of the viscous terms. To further simplify the analysis, we will consier the linearize parabolic problem u t + au x + bu y = c u xx + c 2 u xy + c 2 u yx + c 22 u yy + F, [x, y] Ω, t 0 (3.4) an assume that the bounary ata are homogeneous. (The analysis hols for homogeneous ata, but introuces unnecessary notation. ) We wish to iscretize u xx an u yy with the compact stencil while the mixe erivatives are approximate using D x an D y. The problem lies in eriving sufficient stability conitions for the resulting scheme. We apply the energy metho to (3.4), an with the use of (3.3) we obtain t u 2 = BT + DI + F O, (3.5) where F O = η u + η F (for an arbitrary constant η > 0) an DI = 0 0 w T (C + C T )w x y, (3.6)

5 where C = Compact schemes 25 [ ] [ ] c c 2 ux, w = c 2 c 22 u y enotes the contribution from the issipative terms. Parabolicity requires that x T (C + C T )x 0. (3.7) The Navier-Stokes equations (3.) satisfy the same relation (3.7) with C ij instea of c ij. (See Nörstrom & Svär (2005).) The bounary terms are given by BT = 0 An energy estimate exists for (a + 2α w )u 2 w (a + 2α e )u 2 e y + 0 (b + 2α s )u 2 s (b + 2α n )u 2 n x. (a + 2α w ) 0, (a + 2α e ) 0, (b + 2α s ) 0, (b + 2α n ) 0. (3.8) 3.2. The non-compact formulation A semi-iscretization of (3.4) employing only the first-erivative SBP operator combine with the SAT metho can be written as v t + ad x v + bd x v = c D x D x v + c 2 D x D y v + c 2 D y D x v + c 22 D y D y v + SAT + F. The iscrete version of the bounary conitions (3.3) is given by (3.9) L w v = α w v w + c (D x v) w + c 2 (D y v) w = g w L e v = α e v e + c (D x v) e + c 2 (D y v) e = g e. (3.0) L s v = α s v s + c 22 (D y v) s + c 2 (D x v) s = g s L n v = α n v n + c 22 (D y v) n + c 2 (D x v) n = g n The penalty term in (3.9) is given by SAT = +τ whx e 0 (L w v g w ) + τ e Hx e N (L e v g e ) +τ s Hy (L sv g s ) e 0 + τ n Hy s(l. (3.) sv g n ) e N Lemma 3.. The scheme (3.9) with homogeneous ata has a non-growing solution, if D is a compact first-erivative SBP operator, τ w, s =, τ e, n = an (3.7), (3.8) hol. Proof. Let F = g w, e, s, n = 0. Multiplying (3.9) by vt T H from the left an aing the transpose lea to t v 2 H = 2c v T wh(d x v) w ( τ w ) 2c 2 v T wh(d y v) w ( τ w ) + 2c v T e H(D x v) e ( + τ e ) + 2c 2 v T e H(D y v) e ( + τ e ) 2c 22 v T s H(D y v) s ( τ s ) 2c 2 v T s H(D x v) s ( τ s ) + 2c 22 v T n H(D y v) n ( + τ n ) + 2c 2 v T n H(D x v) n ( + τ n ) + (2τ w α w + a)v T whv w + (2τ e α e a)v T e Hv e + (2τ s α s + b)v T s Hv s + (2τ n α n b)v T n Hv n + DI.

6 26 K. Mattsson, M. Svär & M. Shoeybi The issipative term is given by [ ] T Dx v [ DI = (C + C D y v T ) H ] [ ] Dx v w, (3.2) D y v which exactly mimics (3.6). If τ w, s =, τ e, n = we obtain t v 2 H = (2α w + a)vwhv T w τ e (2α e + a)ve T Hv e. + (2α s + b)vs T Hv s (2α n + b)vn T Hv n + DI This is completely analogous to (3.5). If (3.8) hol we obtain a non-growing energy The compact formulation A semi-iscretization of (3.4) using compact operators an the SAT metho can be written as v t + ad x v + bd x v = c D 2x v + c 2 D x D y v + c 2 D y D x v + c 22 D 2y v + SAT + F. (3.3) The iscrete version of the bounary conitions (3.3) are now (compare with (3.0)) given by L w v = α w v w + c (S x v) w + c 2 (D y v) w = g w L e v = α e v e + c (S x v) e + c 2 (D y v) e = g e. (3.4) L s v = α s v s + c 22 (S y v) s + c 2 (D x v) s = g s L n v = α n v n + c 22 (S y v) n + c 2 (D x v) n = g n The penalty term in (3.3) is given by SAT = +τ whx e 0 ( L w v g w ) + τ e Hx e N ( L e v g e ) +τ s Hy ( L s v g s ) e 0 + τ n Hy. (3.5) s( L s v g n ) e N Lemma 3.2. The scheme (3.3) with homogeneous ata has a non-growing solution, if D is a compact first-erivative SBP operator, D 2 is a compact secon-erivative SBP operator, x T (M Q T H Q)x 0, τ w, s =, τ e, n = an (3.7), (3.8) hol. Proof. Let F = g w, e, s, n = 0. Multiplying (3.3) by vt T H from the left an aing the transpose lea to t v 2 H = 2c v T wh(d x v) w ( τ w ) 2c 2 v T wh(s y v) w ( τ w ) + 2c v T e H(D x v) e ( + τ e ) + 2c 2 v T e H(S y v) e ( + τ e ) 2c 22 v T s H(D y v) s ( τ s ) 2c 2 v T s H(S x v) s ( τ s ) + 2c 22 v T n H(D y v) n ( + τ n ) + 2c 2 v T n H(S x v) n ( + τ n ) + (2τ w α w + a)v T whv w + (2τ e α e a)v T e Hv e + (2τ s α s + b)v T s Hv s + (2τ n α n b)v T n Hv n + DI c In Lemma (3.) it is shown that by using the first erivative twice, i.e., (D ) 2 = H ( Q T H Q+ BD ) to approximate the two terms, c u xx an c 22 u yy, we obtain DI as in (3.2). The issipative part of (D ) 2 is given by Q T H Q. If the compact secon erivative D 2 = H ( M + BS) is use, the issipative part is given by M. We restate M as.

7 Compact schemes 27 M = Q T H Q+(M Q T H Q) an efine the rest term R M Q T H Q. Accoring to the assumption, R is positive semi-efinite. Then we obtain for the compact scheme the issipative term analogous to (3.2), DI c = DI c v T R x H y v c 22 v T H x R y v, which mimics (3.6) with two small aitional amping terms. Finally, for stability of the scheme we also nee to boun the bounary terms. With τ w, s =, τ e, n = we obtain t v 2 H = (2α w + a)v T whv w τ e (2α e + a)v T e Hv e + (2α s + b)v T s Hv s (2α n + b)v T n Hv n + DI c. This is completely analogous to (3.5). If (3.8) hol we obtain a non-growing energy Definiteness of R Consier a pth-orer accurate iscretization of the perioic problem (3.3). One can easily erive the following relations for the rest term R (p) (see Lemma 3.2) R (2) = h3 4 D 4 R (4) = + h5 8 D 6 h7 44 D 8 R (6) = h7 80 D 8 + h9 600 D 0 h 3600 D 2 R (8) = + h9 350 D 0 h 2520 D 2 + h D 4 h D 6, (3.6) where D 2 n = (D + D ) n (3.7) is an approximation of 2 n x ; (D 2 n + D v) j = (v j+ 2v j +v j )/h 2 is the compact seconorer finite ifference approximation. By using Fourier analysis, it is easily shown (see, for example, Mattsson et al. (2004)) that R (p) constitutes only issipative terms. We have shown that the non-compact stencil plus a issipative term is equal to the compact scheme for a Cauchy problem. Since we have erive an energy estimate for the non-compact scheme, we conclue that the same estimate (with an aitional issipative term) will lea to stability for the compact scheme as well. However, a careful bounary closure is require in orer to maintain this property for initial-bounary value problems. Lemma (3.2) introuces a relation between the compact first- an secon-erivative SBP operators via R M Q T H Q. For the secon-orer accurate case we have R (2) = h The following Theorem can be proven (although it is not shown here). 4. (3.8) Theorem 3.3. Let A be an n n pentaiagonal symmetric matrix. Assume n j= A ij =

8 28 K. Mattsson, M. Svär & M. Shoeybi 0 an A ii > 0, A i,i+ < 0, A i,i+2 > 0. If A 2,3 2A 2,4, A n,n 2A n,n 2 an A i,i+ 2A i,i+ + 2A i,i+2, i = 3...n 2, then A is positive semi-efinite. Corollary 3.4. The matrix, R (2), given by (3.8) is positive semi-efinite. Proof. The conitions in Theorem 3.3 can easily be verifie. In the fourth- an sixth-orer cases it is possible to erive similar Theorems. After extensive analysis (not shown here) we conclue that the compact schemes are boune by energy estimates, if the bounary closures are chosen properly. 4. Results We compare the efficiency of the compact an the non-compact formulation by performing numerical simulations of the non-linear problem (3.2). Choosing c = c 2 = c 2 = c 22 = ɛ, we construct the analytic solution ( ) a (( α) x + α y ct) u = a tanh + c, (4.) 2 ɛ which escribes a two-imensional viscous shock. The parameter α efines the propagation angle of the shock, an a,c can be chosen arbitrarily. We solve the problems on a rectangular omain Ω. The stanar explicit fourth-orer Runge-Kutta metho is use for time integration. One of the leaing motives (see Section ) for using a compact formulation was to have amping on the high-frequency moes, which are often triggere by unresolve features in the solution (like a shock). In the first test we compare the compact an non-compact fourth-orer iscretizations with a =, c = 2, an α = 0.2, allowing the viscous shock to travel out through the north-east bounary. The results are shown in Fig. 2. In the secon test we choose a =, c = 0 to obtain a stationary viscous shock. This means that there are no ispersive errors present in the computation, which isolates the issipative errors. To test the efficiency in hanling milly uner-resolve problems (strong shocks require aitional artificial issipation), we compare the secon-orer formulations for the case with ɛ = 0.0. For N < 00, this is a slightly uner-resolve problem. To obtain a solution with an l 2 (error) < 0.0, the compact secon-orer formulation requires 38 2 gri points, an the non-compact 2n-orer formulation requires 94 2 gri points. This is ue to the presence of high-frequency moes in the non-compact formulation. In Fig. 2 we show a comparison of the convergence history between the compact an the non-compact formulation on the mesh with 38 2 gri points. Both solutions were run to t = 0. The compact iscretization is clearly superior. 4.. The two-imensional compressible Navier-Stokes To emonstrate that the stability properties of (3.4) carry over to the 2-D Navier-Stokes equations, we will compute an analytic viscous shock solution (see Svär & Norström (2006)) an laminar flow over a cyliner. (In fact, the the stability properties carry over to the 3-D Navier-Stokes equations as well.) In the first test we chose a computational omain 0 x, y 3. The shock is initiate 0.25 unit lengths away from the iagonal of the box an is propagate with an angle of 45 o, 0.5 length units across the gri. The Reynol number is 0, which results in a smooth profile. A convergence stuy is shown in Table. The convergence rate is calculate as

9 Compact schemes Convergence moving shock, compact v/s non compact, ε = 0., N= log ( l 2 error ) log ( time ) (a) Unsteay shock Convergence steay shock, compact v/s non compact, ε = 0.0, N= log ( l 2 error ) log ( time ) (b) Steay shock Figure 2. The convergence histories. The soli line (an circles) are the fourth-orer compact an the ashe (an boxes) the non-compact.

10 220 K. Mattsson, M. Svär & M. Shoeybi N l (compact) 2 q (compact) l (non compact) 2 q (non compact) Table. log(l 2 error) an convergence rate, q, for the compact an non-compact stencils on Cartesian gris. Drag Maximum Drag Maximum Lift Base Pressure Coefficient Coefficient Coefficient Coefficient Strouhal Separation Recirculation U min number Angle Bubble length Table 2. Simulation results. ( w w (h ) ) ( ) h h q = log 0 / log w w (h 2) 0, (4.2) h h 2 where w is the analytic solution an w (h) the corresponing numerical solution with gri size h. w w (h) h is the iscrete l 2 error. Although the stuy is not shown here, the ifference in accuracy between the compact an wie stencil formulations (for a steay problem) was foun to be larger when the shock is not fully resolve (by increasing the Re number, using a coarse gri). In the secon test we compute the flow over a 2-D cyliner at Re D = 00. An unstructure finite volume coe for compressible N-S equations is use to simulate the flow. The free-stream Mach number is set to 0. to be able to compare the results from previous incompressible simulations. The omian is a cyliner of raius 30 D. Structure gri is use near the cyliner while unstructure gri is use to capture the wake. To resolve the thin laminar bounary layer in front of the cyliner, the minimum raial gri spacing is set to 3 0 3, while 260 points are use in the circumferintial irection. Figure 3 shows the contour of ensity. Table 2 shows the results, which are in goo agreement with the results presente in Kravchenko (998); Kwon & Choi (995).

11 Compact schemes 22 Figure 3. Flow over a 2-D cyliner (ensity), Re D = Conclusions an future work Our approach have been to use SBP operators an the SAT technique to enforce the bounary conitions. By using energy estimates it is proven that there are compact SBP operators that lea to stability for problems with mixe an pure secon erivatives, such as the compressible Navier-Stokes equations. Numerical computations for both Burgers equation an the two-imensional Navier-Stokes equations corroborate the stability properties an also show that the compact schemes are more accurate than the corresponing non-compact schemes. The next step will be to couple the compact unstructure finite volume iscretization to a high-orer finite ifference iscretization to obtain an efficient hybri Navier-Stokes solver. This will allow us to capture the geometry using the unstructure metho an also allow us to capture the wave propagation in the farfiel. Acknowlegments This work is supporte by the Avance Simulation an Computing Program of the Unite States Department of Energy.

12 222 K. Mattsson, M. Svär & M. Shoeybi REFERENCES Carpenter, M. H. & Norström, J. & Gottlieb, D. 999, A Stable an Conservative Interface Treatment of Arbitrary Spatial Accuracy. J. Comp. Phys. 48, Carpenter, M. H. & Gottlieb, D. & Abarbanel, S. 994, Time-stable bounary conitions for finite-ifference schemes solving hyperbolic systems: Methoology an application to high-orer compact schemes. J. Comp. Phys., Gustafsson, B. & Kreiss, H.-O. & Oliger, J. 995, Time epenent problems an ifference methos. Wiley, New York. Kravchenko, A. G. 998, B-spline methos an zonal gris for numerical simulation of turbulent flows Ph.D. Thesis, Deparment of Mechanical Engineering, Stanfor University. Kreiss, H.-O. & Scherer, G. 974, Finite element an finite ifference methos for hyperbolic partial ifferential equations. Mathematical Aspects of Finite Elements in Partial Differential Equations., Acaemic Press, Inc. Kwon, K. & Choi, H. 994, Control of laminar vortex sheing behin a circular cyliner using splitter plates. Phys. Fluis 8(2), Lele, S. K. 992, Compact Finite Difference Schemes with Spectral-like Resolution. J. Comp. Phys. 03, Mattsson, K. & Svär, M. & Norström, J. 2004, Stable an Accurate Artificial Dissipation. J. Sci. Comput. 2, Mattsson, K. & Norström, J. 2004, Summation by parts operators for finite ifference approximations of secon erivatives. J. Comp. Phys. 99, Norström, J. & Svär, M. 2005, Well Pose Bounary Conitions for the Navier- Stokes Equations. SIAM J. Num. Anal. 43, Stran, B. 994, Summation by Parts for Finite Difference Approximations for /x. J. Comp. Phys. 0, Strang, G. 964, Accurate partial ifference methos II. Non-linear problems Num. Math. 6, Svär, M. & Norström, J. 2006, On the orer of accuracy for ifference approximations of initial-bounary value problems. J. Comp. Phys. 28,