A CONSERVATIVE, SECOND ORDER, UNCONDITIONALLY STABLE ARTIFICIAL COMPRESSION METHOD

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1 A CONSERVATIVE, SECOND ORDER, UNCONDITIONALLY STABLE ARTIFICIAL COMPRESSION METHOD VICTOR DECARIA, WILLIAM LAYTON AND MICHAEL MCLAUGHLIN Abstract. This report presents a new artificial compression method for incompressible, viscous flows. The method has second order consistency error and is unconditionally, long time, energy stable for the velocity and, weighted by the timestep, for the pressure. It uncouples the pressure and velocity and requires no artificial pressure boundary conditions. When the viscosity ν = 0 the method also exactly conseves a system energy. The method is based on a Crank-Nicolson Leapfrog time discretization of the slightly compressible model (1 ε 1 grad div)u t + u u + 1 (div u)u ν u + p = f and ε p t + div u = 0. This report presents the method, gives a stability analysis, presents numerical tests and gives a preliminary analysis with tests of the non-physical acoustic waves generated. Consideration of the physical fidelity of the artificial compression method leads to a related method. Key words. artificial compression, Crank-Nicolson, Leapfrog 1. Introduction. Consider the time dependent incompressible Navier-Stokes equations in a d or 3d domain Ω for the fluid velocity and pressure, u(x, t), p(x, t): u t + u u ν u + p = f and div u = 0 in Ω (0, T ], (1.1) p(x, t) dx = 0, u = 0, on Ω and u(x, 0) = u 0 (x). Ω Respectively, ν, f, u 0 are the kinematic viscosity, body force and initial velocity. As problems become larger and assessment of uncertainty becomes necessary, execution time can become of primary importance. Execution time limitations often force the coupling between the velocity and pressure to be broken in various ways including adding a small (artificial) compression term, studied herein. Doing so speeds up the computations dramatically and does not require pressure boundary conditions but introduces extra numerical errors and new physical flow behaviors associated with compressibility, e.g., [Z06]. These include non-physical fast pressure oscillations (acoustics), analyzed in Section 3. These fast acoustic waves can yield restrictive timestep conditions for explicit time discretization of the pressure equation. The method presented below is a second order, artificial compression method with explicit treatment of the pressure that is, nevertheless, unconditionally stable. Since the velocity-pressure uncoupling is through the time discretization and applicable to other space discretizations, we suppress the (secondary) spacial discretization. In the tests in Sections 3 and 4, a standard finite element method is used for spacial discretization. Algorithm 1.1 (Artificial Compression Method). Given time step k > 0, t n = nk, and u n (x) = u(x, t n ), p n (x) = p(x, t n ). Pick α, β > 0 constants with αβ 1 4, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 1560, USA, s: vpd7@pitt.edu, mem66@pitt.edu, wjl@pitt.edu; The research herein was partially supported by NSF grant DMS

2 Let either u n = u n or u n = u n + u n Given (u n, p n ), (u n 1, p n 1 ), find (u n+1, p n+1 ) satisfying: u n 1 + u n 3. (1.) u n+1 u n 1 βk 1 grad div(u n+1 u n 1 )+ ( k ) +u un+1 + u n 1 n + 1 ( ) (div un+1 + u n 1 u n) + ν ( u n+1 + u n 1 ) + p n = f(x, t n ), (1.3) αk(p n+1 p n 1 ) + div u n = 0, u n+1 = 0 on Ω and p n+1 dx = 0. The roles played by the parameters α (units 1/L ), β (units L ) are as follows. αk is the standard artificial compression parameter that allows the pressure to be advanced explicitly in time. The β term is a dispersive regularization that acts through the momentum equation to ensure unconditional stability of the continuity equation, Remark 3.1 below. Thus if β = 0 the method would require a time step condition for stability. The term does increase the condition number and the coupling among velocity components in the linear system to be solved at each step. In d the increased coupling can be corrected using the sparse-grad-div adjustment of [BBR14]. The nonlinearity is explicitly skew symmetrized in the second line of the algorithm and in the numerical tests in Section 4. The extrapolation (1.) is an idea of Ingram [I13] with roots back to Baker [B76]. Often uncoupling of velocity and pressure has been achieved by fractional step or operator splitting methods, e.g., [T69], [Y71], [PT83]. Further efficiency gains per step can be obtained at a cost of an Re related time step condition by explicit discretization of the nonlinear terms; for interesting recent work in this approach see [YBC16]. This method (1.3) is inspired by the original method of Chorin [C68] and a recent pressure correction method connected with the stabilized implicit-explicit (IMEX) method CNLFstab of [JKLMT15]. Artificial (or quasi- or pseudo-) compressibility methods are closely related to pressure projection methods. See Prohl [P97] and Guermond, Minev, Shen [GMS06] for surveys and [A15], [GHKL15], [JL04], [K86],[OST10], [GM15], [GM17] and [OA10] for recent developments in this very active field. Algorithm 1.1 is a time discretization of the slightly compressible model Ω (1 β grad div)u t + u u + 1 (div u)u ν u + p = f, (1.4) αk p t + div u = 0. In all AC methods p(x, 0) must be specified but pressure boundary conditions are not required. In our tests we take the Stokes solution to the given problem data as initial condition. The method (1.3) is an IMEX combination of second order methods (Crank-Nicolson and Leapfrog). The α, β terms are O(k ) perturbations. Thus, the consistency error is O(k ). Theorem.1 (below) shows the velocity is L stable and the pressure weighted by the timestep is L stable: u n + k p n C(data).

3 This stability result, applied to the error equation, suggests the expected rates of convergence of O(k ) and O(k) for the velocity and pressure respectively (e.g., [S96a]). These rates are observed in the tests in Section 4. Section 3 presents preliminary tests on the nonphysical acoustic phenomena introduced by the artificial compression. Section 5 presents conclusions and a few natural open questions.. Unconditional stability. We prove next unconditional, long time, energy stability of the velocity and timestep-weighted pressure in the artificial compression method. For this, let (, ), denote the usual L (Ω) inner product and norm. The proof is based on a time-discrete energy analysis that identifies in (.3) the method s dissipation rate and energy. The first is exactly equal to the physical dissipation which vanishes when ν = 0 The method s kinetic energy, E n+1/, is total dissipation = ν u n+1 + u n 1. E n+1/ := 1 [ u n+1 + u n + β ( div u n+1 + div u n ) (.1) + αk ( p n+1 + p n ) + k(div u n, p n+1 ) k(p n, div u n+1 )]. The energy E n+1/ is quadratic in the flow variables and E n+1/ = E 1/ when ν = f = 0. It has a meaningful interpretation of an energy when positive definite, (.) below, in the flow variables (i.e., when αβ > 1/4). The first term in E n+1/, 1 u, is the standard NSE kinetic energy. The second component αk p + k(div u, p) reflects compressibility effects that vanish as k 0. The β terms in E n+1/ arise from dispersive effects, Remark 3.1 below, and vanish as u 0. Thus when αβ > 1/4, E n+1/ is consistent with the NSE s kinetic energy as k 0 and u 0 and this method, like that of [VVW03], is conservative when αβ > 1/4. Theorem.1 (Unconditional stability). Suppose α > 0, β > 0 are chosen so that We have E n+1/ 1 [ u n+1 + u n + k β αβ > 1 4. ( αβ 1 4) ( pn+1 + p n ) ]. (.) For any timestep k and any t n we have the following energy equality and stability n 1 E n+1/ + k l=0 n 1 E n+1/ + k l=0 ν u l+1 + u l ν u l+1 + u l n 1 = E 1/ + k and thus l=0 n 1 E 1/ + k l=0 (f(t n ), u n+1 + u n 1 ) (.3) 1 ν f(t l+ 1 ) 1 (.4) Proof. Existence of u n+1 follows easily from the estimates used to prove stability and standard linear analysis. We first analyze positivity of E n+1/. To begin, note 3

4 that k(div u n, p n+1 ) k(p n, div u n+1 ) β( div u n+1 + div u n ) + 4β k ( p n+1 + p n ). Thus E n+1/ 1 [ u n+1 + u n + k β ( αβ 1 4) ( pn+1 + p n ) ]. When αβ > 1/4 this is positive definite as claimed. To prove the energy equality, in Algorithm 1.1 take the inner product in the discrete momentum equation with u n+1 + u n 1 and the inner product in the discrete continuity equation with p n+1 + p n 1. Integration by parts and using the explicit skew symmetrization of the nonlinearity gives 1 k [ u n+1 + β div u n+1 ] 1 k [ u n 1 + β div u n 1 ] (p n, div (u n+1 + u n 1 )) + ν u n+1 + u n 1 = (f(t n ), u n+1 + u n 1 ) and αk p n+1 αk p n 1 + (div u n, p n+1 + p n 1 ) = 0. Multiply both equations through by k and add the two resulting equations. yields 1 [ u n+1 + β div u n+1 + αk p n+1 ] 1 [ u n 1 + β div u n 1 + αk p n 1 ] +k(div u n, p n+1 + p n 1 ) k(p n, div (u n+1 + u n 1 )) +kν ( u n+1 + u n 1 ) = k(f(t n ), u n+1 + u n 1 ). Adding and subtracting 1 [ u n + β div u n + αk p n ] gives This 1 [ u n+1 + u n + β ( div u n+1 + div u n ) + αk ( p n+1 + p n ) ] 1 [ u n + u n 1 + β ( div u n + div u n 1 ) + αk ( p n + p n 1 ) ] +k(div u n, p n+1 + p n 1 ) k(p n, div (u n+1 + u n 1 )) (.5) +kν ( u n+1 + u n 1 ) = k(f(t n ), u n+1 + u n 1 ). The terms k(div u n, p n+1 + p n 1 ) k(p n, div (u n+1 + u n 1 )) couple the pressure and incompressibility and are the key terms in the stability analysis. Rearrange them into an exact time difference as follows k(div u n, p n+1 + p n 1 ) k(p n, div (u n+1 + u n 1 )) = = [k(div u n, p n+1 ) k(p n, div u n+1 )] [k(p n, div u n 1 ) k(div u n, p n 1 )]. 4

5 Insert the rearrangement of the coupling terms into (.5). kinetic energy (.3), (.5) becomes Recalling the discrete E n+1/ E n 1/ + kν ( u n+1 + u n 1 ) = k(f(t n ), u n+1 + u n 1 ). Summing yields the claimed energy equality (.3). The energy equality and the Cauchy-Schwarz-Young inequality then imply stability (.4) provided E N+1/ > 0 for u, p 0, 0. Since this holds for αβ > 1/4 stability thus follows. 3. Testing Non-Physical Acoustic Waves. The method (1.3) is a second order discretization of the hyposonic flow model (1.4) (1 β grad div)u t + u u + 1 (div u)u ν u + p = f, (3.1) αk p t + div u = 0. When β = 0 this reduces to a slightly compressible model analyzed by Shen [S9]. Artificial compression methods replace incompressibility (sound speed c = ) with a pressure equation admitting high speed waves with c(k, α, β) as αk 0. The pressure discretization in (1.3) is a two step method and its solution has two modes. One mode converges to the true pressure in the incompressible NSE. The other mode is a spurious mode. The intuition is that this spurious mode may correspond to a non-physical, fast acoustic wave. This section explores these waves for the following test problem. Flow between offset circles. This test problem (from [JL15]) chooses Ω = {(x, y) : x + y r 1 and (x c 1 ) + (y c ) r }, r 1 = 1, r = 0.1, c = (c 1, c ) = ( 1, 0), f(x, y, t) = ( 4y(1 x y ), 4x(1 x y )) T, ν = 10 3, k = 0.01, T = 100, α = 0.5, β = 1.1. with boundary condition u = 0 on both circles. The flow (inspired by variants of Couette flow, [EP00]), stirred by a counterclockwise rotation (with f 0 at the outer circle), rotates about (0, 0) and interacts with the inner circle. This induces a vortex street which re-interacts with the immersed circle creating more complex structures. This flow also exhibits near wall streaks and a central polar vortex that pulsates. Discretize in space using the usual FEM with Taylor-Hood elements, e.g.,[g89], [P83], [Q93], using FreeFEM++[H1] and in time using (1.3). The mesh is a Delaunay mesh with 150 mesh points on the outer circle and 100 mesh points on the inner circle. The values α = 0.5, β = 1.1 were chosen so αβ = 0.75 > 1/4 without precomputations. The value ν = 10 3 is clearly beyond the setting of linear acoustics where the base state is zero velocity, constant pressure and negligible nonlinearity Fluctuations about a rest state. The usual acoustic equation describes small fluctuations about u = 0, p = const for which the quadratic u u term is negligible. In this case (3.1) takes the form (1 β grad div)u t + {negligible terms} ν u + p = f, (3.) αk p t + div u = 0. 5

6 Taking the divergence of the first equation and / t of the second equation, either p t or u (Gresho and Sani [GS00]) may be eliminated. Assuming div f = 0, we obtain that θ = div u and θ = p both satisfy the second order, damped wave equation [ β + 1]θ tt ν θ t (αk ) 1 θ = 0. This equation suggests that nonphysical oscillations in div u and p will be weakly damped by viscosity. Indeed, substituting (θ = div u or θ = p) θ(x, t) = e at φ(x), where φ, λ = Stokes eigenpair, we find that the decay rate a satisfies a νλ 1 + βλ a + λ αk (1 + βλ) = 0. This yields a decaying oscillations of the form θ(x, t) = e At (cos Bt + i sin Bt)φ(x), where A = 1 νλ 1 + βλ and B = 1 λ αk (1 + βλ) ( ) νλ. (3.3) 1 + βλ Sound Speeds. If viscosity is negligible (ν = 0), the sound speed of the nonphysical acoustic wave predicted by the above wave equation will vary with spacial frequency. Indeed, suppose p(x, 0) = e iωx, p t (x, 0) = 0. Setting p(x, t) = e iω(x+ct) and solving for the wave speed c = c(ω) gives c (k, ω) = (αk ) βω. (3.4) As expected, as k 0, the wave speed c which is necessary for consistency with the incompressible limit. This effect is modulated by the fact that higher frequencies in space have slower wave speeds due to the effect of the β terms. Remark 3.1. The effects of the α, β terms. In the artificial compression approximation, u = 0 is replaced by αk p t + u = 0. This introduces nonphysical acoustic waves with wave speed c 1/ αk. In the present method no CFL type timestep restriction is necessary due to the β grad div u t term. This term is a regularization of the problems kinetic energy altering the natural time scales of the problem, e.g., [LR13]. Increasing β slows the nonphysical acoustic waves to the point that the natural CFL condition is satisfied. Physical viscosity slowly damps these waves. Increasing β also slows the rate of damping of viscosity. Acoustic wave-speeds vs. timestep test. To test if wave speeds do indeed increase as the timestep k 0 for the flow between offset circles we computed and plotted (next) the pressure at (x, y) = (0, 0) on a time interval after initial transients pass. 6

7 00 Pressure at Origin vs Tim e, Re = 1000 dt=1/5 dt=1/50 dt=1/100 dt=1/ pressure at origin t Fig. 3.1: Re = 10 3 : Frequency increases as timestep decreases The pressure does indeed oscillate with greater frequency as the timestep decreases consistently with (3.6). Further, for fixed timestep, the amplitude also increases as Re increases for fixed t (but not as t increases for fixed Re). One possible explanation explored in Section 3. is the size of the nonlinear acoustic sound source. Time filters are a promising solution to the problem of nonphysical oscillations in artificial compression methods. An example of their positive effect on this oscillation is presented in Section Nonlinear Acoustics of the PDE Model. The analysis in the last subsection is for perturbation about a rest state u = 0, p = const. For acoustics about a state of motion, the nonlinear term is significant and the analog of the Lighthill model (see, Lighthill [Li5] and [LN09] for justification) is developed as follows. Let the usual Lighthill sound source be denoted Q(u, u) := u : ( u) T = i,j=1,,d u i x j u j x i, for d = or 3. Repeating the above derivation and retaining the nonlinear and viscous terms yields the acoustic equation αk [ β + 1]p tt p = div (u u + 1 ) (div u)u ν u. Lemma 3.. We have div (u u + 1 ) (div u)u = Q(u, u) 1 u (div u) + 1 div u. (3.5) 7

8 Proof. This is a vector identity. Thus if compressibility has negligible effects on acoustic waves then the RHS reduces to the usual Lighthill sound source div (u u + 1 ) (div u)u ν u Q(u, u), if div u 0. For artificial compression methods div u = αk p t 0, then div (u u + 1 ) (div u)u ν u = = Q(u, u) + αk u p t + αk ν p t + α k 4 p t. For ν < k the leading order non-physical source is (formally) αk u p t. Thus, the equation for pressure fluctuations about a non-rest state is αk [ β + 1]p tt p = Q(u, u) + αk u p t αk ν p t + α k 4 p t. (3.6) The nonphysical sound source terms αk u p t - αk ν p t + α k 4 p t do (formally) vanish as αk 0. The question is however what is its size relative to the physical sound source Q(u, u). We test the relative sizes of the Lighthill sound source to the leading order, non-physical sound source in the above, offset circle test problem. Magnitude of the Lighthill vs. non-physical sound source. The test computed the ratio of the Lighthill sound source to the leading order, nonphysical sound source ratio = αk u p t L. Q(u, u) L The time derivative p t was evaluated by second order central time differences. This ratio is plotted below in Figures 3. and 3.3 for k = 1/5, 1/50, 1/100, 1/00. The relative size of the nonphysical acoustic source is small but the ratio undergoes large fluctuations. As the timestep decreases, the ratio becomes more intermittent and the peaks increase. The increase is clearer when the previous figure is compared to the same data with the smallest timestep (1/00) omitted, next Evolution of u. Since u = 0 is replaced by αk p t + u = 0 one natural question is: How close to incompressible is the computed solution? This question is complicated by the (surprising when first discovered) fact that the standard, centered FEM using Hood-Taylor elements yields an approximate solution with u = O(1), [CELR11]. Figure 3.4 presents the evolution of ratio = u / u ; decreasing the timestep decreases the relative size of u. As for the standard FEM, while u decreases as k 0, it is significant for reasonable timestep values. The standard corrections, not tested herein, are either elements with discontinuous pressures or grad div stabilization in the space discretization Galilean Invariance. Comparing (3.1) to the equations of slightly compressible fluids is useful. At small Mach number, the correct model for the pressure equation in isothermal, hyposonic flow in dimensional form is αk (p t + u p) + div u = 0, (3.7) 8

9 αk u p t / Q evolution in time, Re = 1000 dt= 1/5 dt= 1/50 dt= 1/100 dt= 1/00 αk u p t Q t Fig. 3.: The ratio: Non-physical acoustic source / Lighthill source αk u p t / Q evolution in time, Re = 1000 dt= 1/5 dt= 1/50 dt= 1/ αk u p t Q t Fig. 3.3: The same ratio without k = 1/00 data 9

10 u / u evolution in time, Re = 1000 dt= 1/5 dt= 1/50 dt= 1/100 dt= 00 u u t Fig. 3.4: Relative size of div u vs time Zeytounian [Z04], Section.3, or Kreiss [K95] eqn. (1.4). The time derivative p/ t rather than the material derivative dp/dt means (1.4) violates Galilean invariance and is a source of criticism of the physical fidelity of artificial compression methods. Correction requires replacing p/ t with the material derivative. Preserving unconditional stability then requires making the discrete pressure equation linearly implicit as follows. Algorithm 3.3 (Corrected Artificial Compression Scheme). Pick α, β > 0 with αβ > 1/4 and let u n be as in Algorithm 1.1. Given (u n, p n ), (u n 1, p n 1 ), find (u n+1, p n+1 ) satisfying: αk { pn+1 p n 1 k u n+1 u n 1 βk 1 grad div(u n+1 u n 1 )+ ( k ) +u un+1 + u n 1 n + 1 ( ) (div un+1 + u n 1 u n) + ν ( u n+1 + u n 1 ) + p n = f(x, t n ), (3.8) } + div u n = 0, + u n p n+1 + p n (div u n) p n+1 + p n 1 u n+1 = 0 on Ω and p n+1 dx = 0. The error distribution for this method for a test problem with known exact solution is presented in Figure 4.1d. The nonlinear term (u p) in the material derivative dp/dt is explicitly skew symmetrized above. This preserves unconditional stability. 10 Ω

11 Theorem 3.4. Under the same conditions, the above corrected artificial compression scheme satisfies the same stability result as in Theorem.1. Proof. Take the inner product in the discrete momentum equation with u n+1 + u n 1 and the inner product in the discrete continuity equation with p n+1 + p n 1. The momentum equation is the same as previously. The discrete continuity equation gives αk p n+1 αk p n 1 + αk [(u n (p n+1 + p n 1 ), p n+1 + p n 1 ) + 1 ] (div u n (p n+1 + p n 1 ), p n+1 + p n 1 ) +(div u n, p n+1 + p n 1 ) = 0. The bracketed nonlinear terms in the discrete continuity equation satisfy (u n (p n+1 + p n 1 ), p n+1 + p n 1 ) + 1 ((div u n) (p n+1 + p n 1 ), p n+1 + p n 1 ) = 0. Thus, the continuity equation becomes the same as in the proof of Theorem.1 αk p n+1 αk p n 1 + (div u n, p n+1 + p n 1 ) = 0. The remainder of the proof is identical to that of Theorem Time filters control nonphysical acoustics. Time filters are commonly used in geophysical flow calculations to control nonphysical oscillations, e.g., [W10]. We test next the basic Robert-Asselin filter [W10], [HLT14] on the oscillations observed in Figure 3.1. Each timestep, given (u n 1, p n 1 ) and (u n, p n ) Algorithm 1.1 produced (u n+1, p n+1 ). The discrete time-curvatures are the calculated κ u := u n+1 u n + u n 1 and κ p := p n+1 p n + p n 1. The values (u n, p n ) are the updated using the coefficient γ (typically 0 < γ 0. in GFD) by u n u n + γ (un+1 u n + u n 1 ), p n p n + γ (pn+1 p n + p n 1 ). This method is still unconditionally stable provided αβ > 1 4(1 γ). (3.9) Denoting the discrete time-curvatures before and after the filter respectively by κ old,κ new, the time filter has the effect of reducing oscillations by κ new = (1 γ)κ old, see e.g., Section.1 of [HLT14]. To isolate persistent oscillations from initialization, we start the fluid at rest and ramp up the body force. Applying the time filter every step of Algorithm 1.1 for the flow between offset circles, taking k = 1/100, the pressure oscillations in Figure 3.5a are modified to those in Figure 3.5b. Noting the differences in vertical scales in the figures, the amplitude and frequency of the oscillations have been greatly reduced. In the tests in Section 4. the time filter increased the accuracy in the pressure. Any operation applied at every timestep can significantly alter the accuracy and stability of the computed solution. The analysis of the accuracy and stability of the combination of artificial compression plus time filter is underway. 11

12 pressure at origin pressure at origin t (a) AC t (b) AC+RA Fig. 3.5: p(0, 0, t) with and without filter. Note scale difference. 4. Tests of Accuracy. The accuracy tests in this section solve a d evolutionary Stokes problem (ν = 1) with exact solution from [GMS06]. This is discretized in space by a standard finite element method without penalties, stabilizations or other add-ons and using the package FreeFEM++ [H1]. Taylor-Hood elements were used except for one test with the linear-linear (P1/P1) pair which is known to violate the discrete inf-sup condition. Uniform meshes with 70 nodes per side on the boundary were used. The spacial mesh is thus fine relative to the timestep. Thus, the accuracy tests are for errors introduced by the artificial compression method. Let Ω = ( 1, 1). The exact solution from [GMS06] is u(x, y, t) = π sin t(sin πy sin πx, sin πx sin πy) p(x, y, t) = sin t cos πx sin πy. The timestep was chosen large enough that errors were expected to be dominated by timestepping errors. The exact solutions at t = 0 and t = k are used to initialize. We took T final = Pressure Correction and Artificial Compression. We compare the artificial compression (AC) scheme (1.3) to a related pressure correction (PC) method based on time discretization of the modified continuity equation ε p t + div u = 0 and αk p t n Ω = 0. Such methods generally have greater accuracy away from walls than straightforward AC methods but exhibit a boundary layer in the pressure error. Since the term ε p t controls p a smaller value of the grad-div parameter may be chosen without sacrificing unconditional stability. For all methods we have chosen near minimal parameter values consistent with unconditional stability of each: for AC method : α = 0.5, β = 1.1, for AC+RA method : α = 0.53, β = 1.1, γ = 0.1 for PC method : α =

13 The PC method is unconditionally stable and given by u n+1 u n 1 + αk grad div(u n+1 u n 1 ) ν un+1 + u n 1 k +u n un+1 + u n (div un ) un+1 + u n 1 + p n = f n, αk (p n+1 p n 1 ) + u n = 0. (4.1) Both (1.3) and (4.1) were discretized in space by the same finite element method on the same mesh. The pressure error in the AC method is (Figure 4.1a) observed to be distributed across the domain. The error is reduced by adding the RA filter (Figure 4.1c). Figure 4.1d shows similar error to the AC method with an interference pattern. (a) AC (b) PC (c) AC+RA (d) Corrected AC Fig. 4.1: Pressure error fields at final time t = 10, k = For the PC method the pressure error is concentrated along the boundary as expected, Figure 4.1b. The convergence rates (the slopes in the log-log plots in Figures 4. and 4.3 ) of the artificial compression method was O(k ) for velocity, and O(k) for the pressure as forecasted from the stability properties and consistency error. The AC method produced approximations with one extra significant digit of accuracy in the velocity error test. The log-log plot of the pressure errors, next, shows that the AC method s pressure converges as forecasted. In this test the kinetic energy 13

14 10-1 u h u l L / u l L AC, m = Corrected AC, m =.001 AC+RA, m =.10 Pressure Correction, m = k Fig. 4.: AC & PC methods O(k ) for velocity in the nonphysical acoustic modes was small enough as to not influence the overall pressure error in the artificial compression method. The error plateau in the PC method s pressure error is due to the modelled pressure boundary conditions. Indeed, in a further test, not presented herein, the exact pressure was imposed at the boundary for the PC method. The pressure error then no longer plateaued. 4.. The Linear-Linear Element Pair. We repeated the test with P1/P1 elements which do not satisfy the inf-sup condition. Liu, Liu and Pego [LLP10] concluded that pressure regularizations of the forms εp t + div u = 0 and ε p t + div u = 0 require the discrete inf-sup condition for pressure accuracy while those of the forms εp+div u = 0 and ε p+div u = 0 do not. Since the scheme studied herein was not included in their study due to the β term in the the modified momentum equation, we tested the AC method with P1/P1 elements. Velocity convergence plateaus with P1/P1 elements. The pressure diverged for P1/P1 elements and converged (as above) for Taylor-Hood elements, Figures 4.4 and 4.5. Our conclusion is that the discrete inf sup condition is necessary for convergence of both velocity and pressure for the AC method, in agreement with [LLP10]. 5. Conclusions. The artificial compression method presented is based on a two step method and thus shares the features of multi step methods. These features include efficiency for long time, constant time step simulations and the usual difficulties in starting and restarting after adapting the timestep. The first tests herein suggests that the method (1.3) is a promising tool for long time, constant timestep simulations. The method was observed to be second order convergent for the velocity, first order for the pressure and more accurate than a comparable pressure correction method. The method also does not require artificial boundary conditions for the pressure. 14

15 10 0 p h p l L / p l L AC, m = Corrected AC, m = 1.4 AC+RA, m = Pressure Correction, m = k Fig. 4.3: Pressure errors: AC method O(k), PC method shows error plateau Concerning negative features, the P1/P1 test confirmed that a good time discretization does not redeem a bad space discretization. The approximate pressure did exhibit nonphysical acoustic waves. Their wave speed increased as the timestep decreased but decreased as the spacial frequency increased. For problems with significant Reynolds number, the artificial compression method did contribute a nonphysical sound source that was small compared to the physical sound source. This sound source seems to introduce a resonance and the acoustic oscillations increased in amplitude as Re increased (but not as t increased). In preliminary tests time filters reduced significantly non-physical acoustics. The observed effects of the α, β terms are as follows. Since u = 0 is replaced by αk p t + u = 0, the α term introduces non-physical acoustic waves that speed up as k 0 with wave speed c 1/ αk. This is usually expected to introduce a CFL type timestep restriction for stability. In the present method it does not due to the β term. Increasing β slows the nonphysical acoustic waves to the point that the CFD condition is satisfied. Physical viscosity slowly damps these waves. Increasing β also slows the rate of damping of viscosity. Among open problems, there are many promising algorithmic variants such as (following Kobel kov [K0]) including grad div stabilization to the discrete momentum equation and a small amount of damping in the pressure discretization by, e.g., (1 β grad div)u t + u u + 1 (div u)u ν u γ grad div u + p = f, αk ( p n+1 p n 1 ) + αk (p n+1 + p n 1 ) + div u n = 0. k Time filters to suppress the non-physical acoustic mode are presently under care- 15

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