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Uniformly accurate schemes for relaxation approximations to fluid dynamic equations
PERGAMON
Applied Mathematics Letters
Applied Mathematics Letters 16 (2003) 1123-1127
www.elsevier.com/locate/aml
Uniform ly Accurate Schemes for Relaxat ion Approximations to Fluid Dynamic Equations A. KLAR FB Mathematik, TU Darmstadt 64289 Darmstadt, Germany klar@mathematik.
tu-darmstadt
. de
L. PARESCHI Department of Mathematics University of Ferrara 44100 Ferrara, Italy [email protected]
M . SEAID FB Mathematik, TU Darmstadt 64289 Darmstadt, Germany seaidcDmathematik.tu-darmstadt.de (Received and accepted July 2002)
Communicated by P. Markowich relaxation system for the incompressible and compressible Euler and Navier-Stokes Abstract-A equations is considered. A numerical scheme working uniformly in the above limits is constructed using higher-order nonoscillatory upwind discretiaations and higher-order implicit-explicit time discretization. Numerical results are presented for several test cases. @ 2003 Elsevier Ltd. All rights reserved.
Keywords--Relaxation methods, Asymptotic upwind schemes, Projection scheme.
analysis, Fluid dynamic equations, Higher-order
1. INTRODUCTION Kinetic equations yield in the limit for small Knudsen and Mach numbers an approximation of different limit equations like the incompressible and compressible Euler and Navier-Stokes equations. This is achieved by introducing a space-time scaling x -+ X/E, t -+ t/~r+~, CYE [0, l] in the kinetic equation together with a resealing of the mean velocity proportional to the Mach number in the incompressible case. See, for example, [l]. Such a scaled kinetic equation can be viewed as a relaxation system for the fluid dynamic equations. We mention here the recent work by Jin and Slemrod [2], in which a relaxation system This work was supported by DFG Grant KL 1105/g. 0893-9659/03/S - see front matter @ 2003 Elsevier Ltd. All rights reserved. doi: lO.lOlS/SO893-9659(03)00153-8
1124
A. KLAR et al.
that can produce the Euler, Navier-Stokes, as well as the Burnett limit, has been introduced. Relaxation schemes have been used successfully for several of such systems. A large number of kinetic equations with stiff relaxation terms have been considered in fluid dynamic or diffusive limits; see, for example, [3-s]. Moreover, we note that central schemes [6-81 (in nonstaggered version) are closely related to relaxation schemes since both provide Riemann solver free methods. In the present paper, a scheme working uniformly in the compressible and incompressible regime is developed. Nonoscillatory higher-order spatial discretizations and IMEX Runge-Kutta schemes [9] for the time discretization are employed. The nonoscillatory upwind method for the convective part of the relaxation system turns in the limit into higher-order discretizations of the nonlinear convective parts of the fluid dynamic equations. The paper is organized in the following way. A relaxation system is presented in Section 2. Section 3 describes the time and space discretization. Finally, Section 4 contains a numerical investigation of the scheme.
2. A RELAXATION
SYSTEM
Motivated by the considerations in the introduction, we introduce a relaxation system leading to the incompressible and compressible Euler and Navier-Stokes equations in the isothermal case. Density is denoted by p, pressure by p, and flux is given by q = (ql, q2)T = pu, where u = (ul, 2~~)~ is the velocity. r is the relaxation time. Let CYE [0, l] and 1, o=o,
x(a) =
1
0,
a > 0.
The system reads x(a)&p
+ div q = 0,
&q+divO+Vp=O,
(1)
with 0 E BB2x2 and s(q)=;
vq+(vq)T-~V*qI
WI1 =
a2dzq1, b2dyq1 ( a2&q2, b21$q2 > ’
>
)
where a = (a, b) E W$ is chosen according to the local characteristic numerical results). For E --+ 0, equation (1) yields at the leading order
@=
(2)
speeds (see the section on
q@q - 7-+%[q]. 1+ x(ab
For a! = 0, we obtain with p = 1 + p the isothermal Euler equations &p+divq=O, q@q +vp=o. &q + div P
If corrections to order E are taken into consideration, equations
we get the compressible Navier-Stokes
&p+divq=O,
&q+V.(;q
@q
>+ vp
= EV . (TS[d).
1125
Uniformly Accurate Schemes
For (Y = 1, we obtain the incompressible Navier-Stokes equations as limiting system divq = 0, &q+divq@q+Vp=rAq. The Reynolds number is related to the relaxation time by Re = l/r. For 0 < CY< 1, we obtain the incompressible Euler equations divq = 0,
(6)
&q+divq@q+Vp=O. REMARK
1. We note that the relaxation system reads for CY> 0 divq = 0, &q + div 0 + Vp = 0, a,0
+ V[q]
= -&
(7) (0 - q @ q + E+S[qj)
.
Using the vorticity w = V x q E R and 8 = V x 0 E R2 and
V[w]= ( ;I::;Y >) one obtains
Wql =
(
-4ln1 -a,92
&91 &92 > ’
&w+diJe=O,
ate+Va[4 = --& (e-wq+v~q.q+c~-+i-w)
(8)
with q = +[$J]
3. THE
=
-A+
NUMERICAL
= w.
METHOD
For the spatial discretization of the relaxation system, we use the following discretization for the nonstiff advection parts on the left-hand side in (1) and (8), second- and third-order upwind discretizations are used based on MUSCL or EN0 approaches. For a second-order method, we follow the approach in [3]. To obtain a third-order method, we use for the reconstruction step the third-order interpolant from [6]. The stiff parts in (1) and (8) are treated by higher-order centered differences, as in [4,10]; i.e., we use second- and fourth-order centered differences. For the time discretization, a semi-implicit method is used. The nonstiff parts are treated explicitly and the stiff ones implicitly. Second- and third-order methods based on Runge-Kutta IMEX schemes are used to achieve the asymptotic preserving property and to keep the accuracy in the stiff limit.
4. NUMERICAL
RESULTS
AND
EXAMPLES
In this section, we test the above algorithms in two different situations. In the tests, we used the local characteristic speeds as in [6,7] to define a; i.e., a = u. We consider different test cases for incompressible and compressible regimes. Incompressible
Regime
We consider the following periodic problem (see [6]). Let (z:, y) E [0, 2.rr12.The initial conditions are
tanh(i(y-i)), YIP,
IL1(T Y, 0) =
tanh(i(q-y)),
y>n,
1126
A.
KLAR etal.
(a) E = 10e6, t = 8.
(b) E = 10-6, t = 8.
Figure 1. Results for incompressible and E = 10W6).
Euler and Navier-Stokes
case (CY= l/2, a = 1,
and u2(s,y,0)
= Ssin(z),
with b = 0.05 and p = n/15. We use 128 x 128 spatial gridpoints, cx is chosen as cx = l/2 (incompressible Euler limit) and CY= 1 (incompressible Navier-Stokes limit). In both cases, e is chosen as E = 10w6. Figure 1 shows the vorticity at time t = 8 with 7 = 0.01. For comparison of these results with other methods, see, for example, [6]. Compressible
Regimes
We consider the following periodic problem (see [ll]). conditions are p = 1 and U1(T Y, 0) = { -0.5, 0.5, and ~2(“,Y,o>
-0.05 1 y 2 0.05, else,
= { 0.1 sin(4n2), 0,
(a) E = low6 and t = 4. Figure 2. Compressible E = 10-l).
Let .(z, y) E [-0.5, 0.512. The initial
Euler and Navier-Stokes
-0.05 2 y 2 0.05, else.
(b)E=10-1andt=4. equations (c~ = 0, E = 10V6, ahd
Uniformly
Accurate Schemes
1127
We use 128 x 128 spatial gridpoints. Q is chosen equal to 0. E is chosen as .C= 10V6 (compressible Euler limit) and E = 10-l (compressible Navier-Stokes limit). Figure 2 shows”the evolution of marker particles at the interface at time t = 4. T is chosen as 7 = 0.01 as before.
REFERENCES 1. C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations: Formal derivations, J. Stat. Phys. 63, 323-344, (1991). 2. S. Jin and M. Slemrod, Flegularization of the Burnett equations via relaxation, J. Statist. Phys. 103, 10091033, (2001). 3. S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48, 235-276, (1995). 4. S. Jin, L. Pareschi and G. Toscani, Diffusive relaxation schemes for discrete-velocity kinetic equations, SIAM J. Num. Anal. ‘35, 2405-2439, (1998). 5. A. Klar, An asymptotic-induced numerical scheme for kinetic equations in the low Mach number limit, SIAM J. Num. Anal. 36, 1507-1527, (1999). 6. A. Kurganov and D. Levy, A third-order semi-discrete central scheme for conservation laws and convection diffusion equations, SIAM Journal on Scientific Computing 22, 1461-1488, (2000). 7. A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Camp. Phys. 160, 241-282, (2000). 8 R. Kupferman and E. Tadmor, A fast high-resolution second-order central scheme for incompressible flows, Proc. Nat. Acad. Sci. 94, 4848, (1997). 9. U. Ascher, S. Ruuth and Fl. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Namer. Math. 25, 151-167, (1997). 10 M.K. Banda, A. Klar, L. Pareschi and M. Seaid, Lattic+Boltzmann type relaxation systems and high order relaxation schemes for the incompressible Navier-Stokes equation, Preprint, (2001). 11 C.D. Munz and L. Schmidt, Numerical simulation of compressible hydrodynamic instabilities with high resolution schemes, In Nonlinear Hyperbolic Equations, Notes on Numerical Fluid Dynamics 24, (Edited by J. Ballman and R. Jeltsch), pp. 456-465, (1988).
Applied Mathematics Letters
Applied Mathematics Letters 16 (2003) 1123-1127
www.elsevier.com/locate/aml
Uniform ly Accurate Schemes for Relaxat ion Approximations to Fluid Dynamic Equations A. KLAR FB Mathematik, TU Darmstadt 64289 Darmstadt, Germany klar@mathematik.
tu-darmstadt
. de
L. PARESCHI Department of Mathematics University of Ferrara 44100 Ferrara, Italy [email protected]
M . SEAID FB Mathematik, TU Darmstadt 64289 Darmstadt, Germany seaidcDmathematik.tu-darmstadt.de (Received and accepted July 2002)
Communicated by P. Markowich relaxation system for the incompressible and compressible Euler and Navier-Stokes Abstract-A equations is considered. A numerical scheme working uniformly in the above limits is constructed using higher-order nonoscillatory upwind discretiaations and higher-order implicit-explicit time discretization. Numerical results are presented for several test cases. @ 2003 Elsevier Ltd. All rights reserved.
Keywords--Relaxation methods, Asymptotic upwind schemes, Projection scheme.
analysis, Fluid dynamic equations, Higher-order
1. INTRODUCTION Kinetic equations yield in the limit for small Knudsen and Mach numbers an approximation of different limit equations like the incompressible and compressible Euler and Navier-Stokes equations. This is achieved by introducing a space-time scaling x -+ X/E, t -+ t/~r+~, CYE [0, l] in the kinetic equation together with a resealing of the mean velocity proportional to the Mach number in the incompressible case. See, for example, [l]. Such a scaled kinetic equation can be viewed as a relaxation system for the fluid dynamic equations. We mention here the recent work by Jin and Slemrod [2], in which a relaxation system This work was supported by DFG Grant KL 1105/g. 0893-9659/03/S - see front matter @ 2003 Elsevier Ltd. All rights reserved. doi: lO.lOlS/SO893-9659(03)00153-8
1124
A. KLAR et al.
that can produce the Euler, Navier-Stokes, as well as the Burnett limit, has been introduced. Relaxation schemes have been used successfully for several of such systems. A large number of kinetic equations with stiff relaxation terms have been considered in fluid dynamic or diffusive limits; see, for example, [3-s]. Moreover, we note that central schemes [6-81 (in nonstaggered version) are closely related to relaxation schemes since both provide Riemann solver free methods. In the present paper, a scheme working uniformly in the compressible and incompressible regime is developed. Nonoscillatory higher-order spatial discretizations and IMEX Runge-Kutta schemes [9] for the time discretization are employed. The nonoscillatory upwind method for the convective part of the relaxation system turns in the limit into higher-order discretizations of the nonlinear convective parts of the fluid dynamic equations. The paper is organized in the following way. A relaxation system is presented in Section 2. Section 3 describes the time and space discretization. Finally, Section 4 contains a numerical investigation of the scheme.
2. A RELAXATION
SYSTEM
Motivated by the considerations in the introduction, we introduce a relaxation system leading to the incompressible and compressible Euler and Navier-Stokes equations in the isothermal case. Density is denoted by p, pressure by p, and flux is given by q = (ql, q2)T = pu, where u = (ul, 2~~)~ is the velocity. r is the relaxation time. Let CYE [0, l] and 1, o=o,
x(a) =
1
0,
a > 0.
The system reads x(a)&p
+ div q = 0,
&q+divO+Vp=O,
(1)
with 0 E BB2x2 and s(q)=;
vq+(vq)T-~V*qI
WI1 =
a2dzq1, b2dyq1 ( a2&q2, b21$q2 > ’
>
)
where a = (a, b) E W$ is chosen according to the local characteristic numerical results). For E --+ 0, equation (1) yields at the leading order
@=
(2)
speeds (see the section on
q@q - 7-+%[q]. 1+ x(ab
For a! = 0, we obtain with p = 1 + p the isothermal Euler equations &p+divq=O, q@q +vp=o. &q + div P
If corrections to order E are taken into consideration, equations
we get the compressible Navier-Stokes
&p+divq=O,
&q+V.(;q
@q
>+ vp
= EV . (TS[d).
1125
Uniformly Accurate Schemes
For (Y = 1, we obtain the incompressible Navier-Stokes equations as limiting system divq = 0, &q+divq@q+Vp=rAq. The Reynolds number is related to the relaxation time by Re = l/r. For 0 < CY< 1, we obtain the incompressible Euler equations divq = 0,
(6)
&q+divq@q+Vp=O. REMARK
1. We note that the relaxation system reads for CY> 0 divq = 0, &q + div 0 + Vp = 0, a,0
+ V[q]
= -&
(7) (0 - q @ q + E+S[qj)
.
Using the vorticity w = V x q E R and 8 = V x 0 E R2 and
V[w]= ( ;I::;Y >) one obtains
Wql =
(
-4ln1 -a,92
&91 &92 > ’
&w+diJe=O,
ate+Va[4 = --& (e-wq+v~q.q+c~-+i-w)
(8)
with q = +[$J]
3. THE
=
-A+
NUMERICAL
= w.
METHOD
For the spatial discretization of the relaxation system, we use the following discretization for the nonstiff advection parts on the left-hand side in (1) and (8), second- and third-order upwind discretizations are used based on MUSCL or EN0 approaches. For a second-order method, we follow the approach in [3]. To obtain a third-order method, we use for the reconstruction step the third-order interpolant from [6]. The stiff parts in (1) and (8) are treated by higher-order centered differences, as in [4,10]; i.e., we use second- and fourth-order centered differences. For the time discretization, a semi-implicit method is used. The nonstiff parts are treated explicitly and the stiff ones implicitly. Second- and third-order methods based on Runge-Kutta IMEX schemes are used to achieve the asymptotic preserving property and to keep the accuracy in the stiff limit.
4. NUMERICAL
RESULTS
AND
EXAMPLES
In this section, we test the above algorithms in two different situations. In the tests, we used the local characteristic speeds as in [6,7] to define a; i.e., a = u. We consider different test cases for incompressible and compressible regimes. Incompressible
Regime
We consider the following periodic problem (see [6]). Let (z:, y) E [0, 2.rr12.The initial conditions are
tanh(i(y-i)), YIP,
IL1(T Y, 0) =
tanh(i(q-y)),
y>n,
1126
A.
KLAR etal.
(a) E = 10e6, t = 8.
(b) E = 10-6, t = 8.
Figure 1. Results for incompressible and E = 10W6).
Euler and Navier-Stokes
case (CY= l/2, a = 1,
and u2(s,y,0)
= Ssin(z),
with b = 0.05 and p = n/15. We use 128 x 128 spatial gridpoints, cx is chosen as cx = l/2 (incompressible Euler limit) and CY= 1 (incompressible Navier-Stokes limit). In both cases, e is chosen as E = 10w6. Figure 1 shows the vorticity at time t = 8 with 7 = 0.01. For comparison of these results with other methods, see, for example, [6]. Compressible
Regimes
We consider the following periodic problem (see [ll]). conditions are p = 1 and U1(T Y, 0) = { -0.5, 0.5, and ~2(“,Y,o>
-0.05 1 y 2 0.05, else,
= { 0.1 sin(4n2), 0,
(a) E = low6 and t = 4. Figure 2. Compressible E = 10-l).
Let .(z, y) E [-0.5, 0.512. The initial
Euler and Navier-Stokes
-0.05 2 y 2 0.05, else.
(b)E=10-1andt=4. equations (c~ = 0, E = 10V6, ahd
Uniformly
Accurate Schemes
1127
We use 128 x 128 spatial gridpoints. Q is chosen equal to 0. E is chosen as .C= 10V6 (compressible Euler limit) and E = 10-l (compressible Navier-Stokes limit). Figure 2 shows”the evolution of marker particles at the interface at time t = 4. T is chosen as 7 = 0.01 as before.
REFERENCES 1. C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations: Formal derivations, J. Stat. Phys. 63, 323-344, (1991). 2. S. Jin and M. Slemrod, Flegularization of the Burnett equations via relaxation, J. Statist. Phys. 103, 10091033, (2001). 3. S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48, 235-276, (1995). 4. S. Jin, L. Pareschi and G. Toscani, Diffusive relaxation schemes for discrete-velocity kinetic equations, SIAM J. Num. Anal. ‘35, 2405-2439, (1998). 5. A. Klar, An asymptotic-induced numerical scheme for kinetic equations in the low Mach number limit, SIAM J. Num. Anal. 36, 1507-1527, (1999). 6. A. Kurganov and D. Levy, A third-order semi-discrete central scheme for conservation laws and convection diffusion equations, SIAM Journal on Scientific Computing 22, 1461-1488, (2000). 7. A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Camp. Phys. 160, 241-282, (2000). 8 R. Kupferman and E. Tadmor, A fast high-resolution second-order central scheme for incompressible flows, Proc. Nat. Acad. Sci. 94, 4848, (1997). 9. U. Ascher, S. Ruuth and Fl. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Namer. Math. 25, 151-167, (1997). 10 M.K. Banda, A. Klar, L. Pareschi and M. Seaid, Lattic+Boltzmann type relaxation systems and high order relaxation schemes for the incompressible Navier-Stokes equation, Preprint, (2001). 11 C.D. Munz and L. Schmidt, Numerical simulation of compressible hydrodynamic instabilities with high resolution schemes, In Nonlinear Hyperbolic Equations, Notes on Numerical Fluid Dynamics 24, (Edited by J. Ballman and R. Jeltsch), pp. 456-465, (1988).