A MUSCL-Hancock-ADER scheme for ADRE

Transcrição

1 A MUSCL-Hancock-ADER scheme for ADRE May 016 S. Busto*, E.F. Toro**, M.E. Vázquez-Cendón* *Universidade de Santiago de Compostela, **Università di Trento

2 Outline 1 Motivation ADER 3 Stability, truncation error and monotonicity 4 Numerical results

3 Motivation Motivation Navier-Stokes momentum equation ρu t + div (F (u, ρ)) = divτ gradπ + f u t q(x, t) + x f (q(x, t)) = x (α(x, t) x q(x, t)) + βq(x, t) f (q) = λq, λ R +, β R Advection-diusion-reaction equation

4 Motivation Finite volume methods Finite volume methods t n+1 V x i 1 x i x i+ 1 t n xi+ 1 x i 1 ( ) t n+1 xi+ 1 t q + x f (q) dx dt = x (α x q) + βq dx dt; t n x i 1 t n ( ) t n+1

5 Motivation Finite volume methods Finite volume methods t n+1 t n+1 V V x i 1 x i x i+ 1 t n x i 1 x i x i+ 1 t n xi+ 1 x i 1 ( ) t n+1 xi+ 1 t q + x f (q) dx dt = x (α x q) + βq dx dt; t n x i 1 t n ( ) t n+1 1 xi+ 1 ( q(x, t n+1 ) q(x, t n ) ) dx + 1 x x i x 1 = 1 x xi+ 1 x i 1 ( t n+1 t n x (α x q) dx ) t n+1 t n ( ) f (q(x i+ 1, t)) f (q(x i 1, t)) dt dt + 1 xi+ 1 x x i 1 ( t n+1 t n βq dx ) dt.

6 Motivation Finite volume methods We denote q n+1 i 1 x xi+ 1 x i 1 ( q(x, t n+1 ) ) dx, q n i 1 x xi+ 1 x i 1 (q(x, t n )) dx, f n i+ 1 1 t t n+1 t n ( ) f (q(x i+ 1, t)) dt, f n i 1 1 t t n+1 gi n 1 ( xi+ 1 t n+1 x (α x q) dx t x x i 1 t n si n 1 ( xi+ 1 ) t n+1 βq dx dt. t x x i 1 t n t n ) ( ) f (q(x i 1, t)) dt, dt, Hence, the scheme results q n+1 i = qi n t ( ) f n i+ x 1 f n i 1 + tgi n + tsi n.

7 ADER ADER ADER was rst put forward in : E. F. Toro, R. C. Millington, and L. A. M. Nejad. Towards Very High Order Godunov Schemes, 001. The method is divided into four steps 1 : Step 1. Polynomial reconstruction. Step. Resolution of a generalized Riemann problem. Step 3. Approximation of the diusion term. Step 4. Calculation of the numerical source. 1 S. Busto, E.F. Toro, M.E. Vázquez-Cendón. An extension of ADER Schemes for advection-diusion-reaction equations.

8 ADER Step 1. Polynomial reconstruction Step 1. Polynomial reconstruction We consider a reconstruction of the data in terms of rst degree polynomials: p i (x) = qi n + i (x x i ), p i+1 (x) = q n i+1 + i+1 (x x i+1 ), where i, i+1 are the centred slopes related to the cells i and i + 1. p i(x) x i 3 x i 1 x i 1 x i+ 3 x i x i+ 3 x i+1 i = q i+1 q i 1 x

9 ADER Step 1. Polynomial reconstruction t n+1 t n+1 V V x i 1 x i x i+ 1 t n x i 1 x i x i+ 1 t n ( ) q n+1 i = qi n t x f n f n + tg i+ 1 i 1 i n + tsi n.????

10 ADER. Resolution of the generalized Riemann problem Step. Resolution of the generalized Riemann problem To compute the numerical ux we consider the following GRP: t q(x, t) + λ x q(x, t) = x (α(x, t) x q(x, t)) + βq(x, t), p i (x), x < 0, q(x, 0) = p i+1 (x), x > 0. Generalizations Piecewise polynomia initial condition. Source term. Diusion term.

11 ADER We consider the Taylor series expansion in time. Resolution of the generalized Riemann problem q i+ 1 (τ) = q(0, 0 + ) + τ t q(0, 0 + ), τ = 1 t, then, the solution is divided into a leading term, q(0, 0 + ), and a high order term t q(0, 0 + ). The leading term is computed through a RP: t q(x, t) + λ x q(x, t) = 0, qi n + 1 q(x, 0) = x i, x < 0, q n 1 x i+1 i+1, x > 0, considering x t = 0, q(0, 0 + ) = d (0) (0) = i+ 1 q n i + 1 x i, λ > 0, q n i+1 1 x i + 1, λ < 0. (1) ()

12 ADER Following Cauchy-Kovalevskaya procedure,. Resolution of the generalized Riemann problem t q(x, t) + λ x q(x, t) = βq(x, t) t q(x, t) = λ x q(x, t) + βq(x, t) we can express the time derivative in terms od spatial derivatives: q i+ 1 (τ) = q(0, 0 + ) + τ [ λ x q(0, 0 + ) + βq(0, 0 + )]. (3) Considering the evolution equation for the spacial derivative, t ( x q(x, t)) + λ x ( x q(x, t)) = β x q(x, t), and neglecting the source term t ( x q(x, t)) + λ x ( x q(x, t)) = 0, i, x < λ, x q(x, 0) = i+1, x > λ, it results x q(0, 0 + ) = d (1) (0) = i+ 1 i, λ > 0, i+1, λ < 0. (4)

13 ADER. Resolution of the generalized Riemann problem Finally, taking into account (), (3) and (4) it results q i+ 1 (τ) = q(0, 0 + ) + τ t q(0, 0 + ) qi n + 1 x i + τ [ λ i + β ( qi n + 1 x )] i = q n 1 x i+1 i+1 + τ [ λ i+1 + β ( q n 1 x )] i+1 i+1 λ > 0, λ < 0. Performing exact integration, the numerical ux reads λ { qi n + 1 x i + τ [ λ i + β ( qi n + 1 x )]} i λ > 0, f i+ 1 = λ { q n 1 x i+1 i+1 + τ [ λ i+1 + β ( q n 1 x i+1 i+1)]} λ < 0.

14 ADER. Resolution of the generalized Riemann problem Numerical ux with non-linear diusion The diusion term is included into the Taylor expansion: q i+ 1 (τ) = q(0, 0 + ) + τ [ λ x q(0, 0 + ) + x (α(0, 0 + ) x q(0, 0 + )) + βq(0, 0 + )] We solve the Riemann problem related to the second derivative: t ( x α(x, t) x q(x, t)) + λ x ( x α(x, t) x q(x, t)) = 0, ( α ) x (α(x, 0) x q(x, 0)) = i, x < λ, ( α ) i+1, x > λ, (5) where ( α ) i denotes the centred nite dierences approximation of second derivative of q in the i th node, x (α(x i, t) x q(x i, t)).

15 ADER. Resolution of the generalized Riemann problem Hence, the value of the conservative variable in the middle of the control volume is = q i+ 1 (τ) = q(0, 0 + ) + τ t q(0, 0 + ) qi n + 1 x i + τ [ λ i + β ( qi n + 1 x ) ] i + ( α )i λ > 0, q n 1 x i+1 i+1 + τ [ λ i+1 + β ( q n 1 x ) ] i+1 i+1 + ( α )i+1 λ < 0. Integrating, we obtain the numerical ux: { λ qi n + 1 x i + τ f i+ 1 = [ λ i + β ( q n i + 1 x i ) ]} () + α i { [ λ q n 1 x i+1 i+1 + τ λ i+1 + β ( q n 1 x i+1 i+1 λ > 0, ]} ) + α () i+1 λ < 0.

16 ADER. Resolution of the generalized Riemann problem t n+1 t n+1 V V x i 1 x i x i+ 1 t n x i 1 x i x i+ 1 t n ( ) q n+1 i = qi n t x f n f n + tg i+ 1 i 1 i n + tsi n.??

17 ADER Step 3. Approximation of the diusion term Step 3. Approximation of the diusion term We consider a centred scheme to compute the diusive term: x (α(x i, t n ) x q(x i, t n )) ( α ) i = α n i+ 1 = 1 ( [α n x i+ q n 1 i+1 qi n = 1 {[ α n x i+ 1 + t tα i+ 1 ) α n i 1 q n i+1 qn i α n x i 1 ( q n i qi 1) ] n q n i q n i 1 x ] [ q n i+1 qi n + t ( ( λ n i+1 n ) i + ( α ) n i+1 ( α )n i + β ( q n i+1 qi n ))] ] [ q n i 1 qi n + t ( ( λ n i 1 n ) i + ( α ) n i 1 ( α )n i + β ( q n i 1 qi n ))]}. [ + α n i 1 + t tα i 1 Remark Taylor expansion in time of the diusive term is a requirement to get a second order scheme.

18 ADER Step 4. Computation of the numerical source t n+1 t n+1 V V x i 1 x i x i+ 1 t n x i 1 x i x i+ 1 t n ( ) q n+1 i = qi n t x f n f n + tg i+ 1 i 1 i n + tsi n.?

19 ADER Step 4. Computation of the numerical source Step 4. Computation of the numerical source Assuming q n i = q n i + τ ( λ i + βq n i ), the numerical source reads s n i = βq n i = β [q n i + τ ( λ i + βq n i )]. Finally, the numerical scheme for λ > 0 results q n+1 i =qi n λ t { qi n + 1 [ ( x x i + τ λ i + β qi n + 1 )] x i q n i [ ( x i 1 τ λ i 1 + β q n i 1 1 )]} x i 1 [ + β t qi n + t ] ( λ i + βqi n ). (6)

20 ADER Step 4. Computation of the numerical source Taking into account diusive terms, [ s i = βqi n = β qi n + t ] ( λ i + ( α ) i + βqi n ). Then, the complete scheme for λ > 0 is q n+1 i =qi n λ t { qi n + 1 [ ( x x i + τ λ i + β qi n + 1 ) ] x i + ( α ) i q n i 1 1 [ ( x i 1 τ λ i 1 + β q n i ) ]} x i 1 + ( α ) i 1 t {[ α n x i+ 1 + t ] [ tα i+ 1 q n i+1 qi n + t ( ( λ n i+1 n ) i + ( α ) n i+1 ( α )n i + β ( q n i+1 qi n ))] [ + α n i 1 + t ] [ tα i 1 q n i 1 qi n + t ( ( λ n i 1 n ) i + ( α ) n i 1 ( α )n i + β ( q n i 1 qi n ))]} [ + β q i + t ] ( λ i + ( α ) i + βq i ). (7)

21 Stability, truncation error and monotonicity Truncation error, monotonicity and stability Truncation error 1 Theorem ADER schemes for advection-reaction equation, (6), and advection-reaction-diusion equation, (7), are second order in time and space. Monotonicity The developed schemes are non-monotone. The coupling between ADER and ENO, WENO, TVD or WAF has been successfully applied for the linear advection equation. Stability Admissible values for c = λ t x, d = α t x and r = β t can be set. 1 S. Busto, E.F. Toro and M.E. Vázquez-Cendón. Design and analysis of ADER-type schemes for model advection-diusion-reaction equations. Preprint submitted to J. Comp. Phys. (016).

22 Stability, truncation error and monotonicity Stability analysis for LAE Stability analysis We follow von Neumann stability analysis procedure: Given a trial function, qj n = A n e iθj, θ = P x, a scheme is stable if and only if it is veried A < 1 θ [ π, π]. Lineal advection equation We substitute the test function on the ADER scheme for the linear advection equation, [ A n+1 e iθj = A n e iθj c A n e iθj A n e iθ(j 1) + 1 c ( A n e iθ(j 1) 4 A n e iθj A n e iθ(j 1) + A n e iθ(j ))], then, the amplication factor reads [ A = 1 c 1 e iθ + 1 c ( e iθ 1 e iθ + e iθ)]. 4

23 Stability, truncation error and monotonicity Lineal advection equation Moreover, we can compute A : A = 1 c cos θ c + 1 c cos 3 θ c 3 cos 3 θ + 1 c 4 cos θ + 1 c 4 cos 3 θ + c cos θ + 3 c c c 4 c cos θ 5 c cos θ + c 3 cos θ 1 c 4 cos θ + 1 = c (c 1) (cos θ 1) [ 1 c (c 1) (cos θ + 1) + 1 ] + 1, then, A 1 c (c 1) (cos θ + 1) c 1 c (c 1) (cos θ + 1) c > 1 Hence, as we have already seen graphically, the scheme is stable for c [0, 1].

24 Stability, truncation error and monotonicity Lineal advection equation To plot the amplication factor, we consider the related functions A c : [ π, π] C Im 1.5 θ A c (θ) = A(θ, c) c=0.5 c=1 c=1.5 c= Amplication factor for set values of c and θ [ π, π]. Re

25 Stability, truncation error and monotonicity Lineal advection equation c 0.8 On the other hand, the denition of A : [ π, π] R R (θ, c) A(θ, c). (8) allows for the study of contour lines. 1.5 A(c,θ) Contours of A(c,θ) c 0 4 θ θ E.F. Toro and S.J. Billett. Centred TVD schemes for hyperbolic conservation laws. In: IMA Journal of Numerical Analysis 0.1 (000), pp

26 Stability, truncation error and monotonicity Linear advection-diusion-reaction equation Linear advection-diusion-reaction equation The amplication factor of the linear advection-diusion-reaction equation reads { A =1 c 1 cos θ + i sin θ + 1 c (i sin θ 1 + cos (θ) i sin (θ)) 4 + r [1 cos θ + i sin θ + 14 ] (i sin θ 1 + cos (θ) i sin (θ)) + d } (4 cos θ i sin θ 3 cos (θ i sin (θ))) { + d cos θ c 4 (i sin (θ) 6i sin θ) + d ( cos (θ) 8 cos θ + 6) + r } { ( cos θ ) + r 1 c i sin θ + d (cos θ 1) + r }. Due to its complexity, we only carry out the graphical stability analysis.

27 Stability, truncation error and monotonicity Linear advection-diusion-reaction equation LARE: c [0, 3], d = 0, r [, 0] a a) b c b) c)

28 Stability, truncation error and monotonicity Linear advection-diusion-reaction equation O c,r = { (θ, c, r) θ [ π, π], c [0, c M ], r [r m, 0], c M R +, r m R }

29 Stability, truncation error and monotonicity Linear advection-diusion-reaction equation LADE: c [0, 3], d [0, 1.5], r = 0 a b c b) c)

30 Stability, truncation error and monotonicity Linear advection-diusion-reaction equation LADRE: c [0, 3], d [0, 1.5], r = 0.5 a) a b c b) c)

31 Stability, truncation error and monotonicity Linear advection-diusion-reaction equation LADRE: c [0, 3], d [0, 1.5], r = 1 a) a b c b) c)

32 Stability, truncation error and monotonicity Linear advection-diusion-reaction equation LAE: O c = { (θ, c) θ [ π, π], c [0, c M ], c M R +} LARE: O c,r = {(θ, c, r) θ [ π, π], c [0, c M ], r [r m, 0], c M R +, r m R } LADRE: O c,d,r = {(θ, c, r, d) θ [ π, π], c [0, c M ], d [0, d M ], r [r m, 0], c M, d M R +, r m R } Remark The time step is computed by setting the bound of c, d and r and computing the maximum t verifying all the constraints: { cm x t = min λ, d M x, r m α β }.

33 Numerical results Numerical results We consider the norms ErrL 1 = q ˆq l (L 1 (Ω)), Err L = q ˆq l (L (Ω)), ErrL = q ˆq l (L (Ω)) with ˆq the computed solution and q the exact or reference solution. And we denote the related orders of accuracy O L 1, O L, O L.

34 Numerical results TEST 1: Advection-reaction problem Advection-reaction problem: t q(x, t) + x q(x, t) = q(x, t), q(x, 0) = exp( x ). Exact solution: q(x, t) = exp( (x λt) + βt). Computational domain: Ω T = [0, ] [0, 1]. Boundary conditions: c M = 1, r m = 1. Dirichlet.

35 Numerical results nodes 51 nodes Exact solution x

36 Numerical results Cells ErrL 1 O L 1 Err L O L Err L O L r 8.15E.17E.95E E E E E E E E E E E E E E E E E E E Test 1. Columns from second to seventh show the errors and convergence rates obtained. The last column depicts the values computed for r = β t = t. Ω = [0, ], t end = 1, c = λ t = c x M = 1.

37 Numerical results TEST : Advection-reaction problem with piecewise soution Generalized Riemann problem: t q(x, t) + 1 xq(x, t) = q(x, t), 1 x [ 1 q(x, 0) =, 8 ] 1, 0 x [ ( 0, 8) 1 1, 1.5], Exact solution: 1 x 1 q(x, t) = t [ 1, 8 ] 1, 0 x 1t [ ( 0, 1 8) 1, 1.5]. Computational domain: Ω T = [0, 1] [0, 1]. Boundary conditions: c M = 0.5, r m = 1. Dirichlet.

38 Numerical results nodes Exact solution x

39 TEST 3: Diusion problem Numerical results Non-linear diusion problem: t q(x, t) = x [ cosh(t + ) sin ( (x 1) ) sinh(t + ) q(x, 0) = x q(x, t) ] sinh() cosh() sin ( (x 1) ), Exact solution: q(x, t) = sinh(t + ) cosh(t + ) sin ( (x 1) ) Computational domain: Ω T = [0, π] [0, 1]. Boundary conditions: d M = 0.5 Periodic ( π ).

40 Numerical results nodes 51 nodes Exact solution x

41 Numerical results Cells ErrL 1 O L 1 Err L O L Err L O L E E E E E E E E E E E E E E E E E E E E E Test 3. Errors and convergence rates obtained. Ω = [ π, π], t end = 1, d M = 0.5.

42 Numerical results Test 4: Advection-diusion-reaction problem Advection-diusion-reaction problem: [ t q(x, t) + 10 x q(x, t) 10 4 x exp(x(t 1) ) x q(x, t) ] = 5q(x, t), q(x, 0) = exp(sin (x)) Reference solution. Computational domain: Ω T = [0, π] [0, 1]. Boundary conditions: Periodic (π). c M = 0.5, d M = 0.5, r m = 0.5.

43 Numerical results Reference solution x

44 Numerical results Cells ErrL 1 O L 1 Err L O L Err L O L E E E E E E E E E E E E E E E E E E Test 4. Errors and convergence rates obtained. Ω = [0, π], t end = 1, c M = 0.5, d M = 0.5, r m = 0.5.

45 Numerical results Cells c d r E E E 6 9.8E E 6 4.9E E 6.45E E 5 1.3E E E 3 Test 4. Values of the parameters c, d and r for the computed time step. Ω = [0, π], t end = 1, c M = 0.5, d M = 0.5, r m = 0.5.

46 Conclusiones Conclusions ADER methodology was extended to solve hyperbolic equations with diusion and reaction terms. Linear stability analysis was performed. Truncation error analysis prove second order in space and time scheme. Empirical convergence rate studies conrm the expected theoretical accuracy.

47 Acknowledgements Acknowledgements This project was partially supported by Spanish MECD under grant FPU13/0079; by Consellería de Cultura Educación e Ordenación Universitaria, Xunta de Galicia, under grant Axudas de apoio á etapa predoutoral do Plan IC; by Spanish MICINN project MTM R; by Xunta de Galicia and FEDER under project GRC and by Fundación Barrié under grant Becas de posgrado en el extranjero.

48 Thank you! S. Busto