Dissipative numerical methods for the Hunter-Saxton equation

Transcription

1 Dissipative numerical methos for the Hunter-Saton equation Yan Xu an Chi-Wang Shu Abstract In this paper, we present further evelopment of the local iscontinuous Galerkin (LDG) metho esigne in [] an a new issipative iscontinuous Galerkin (DG) metho for the Hunter-Saton equation. The numerical flues for the LDG an DG methos in this paper are base on the upwining principle. The resulting schemes provie aitional energy issipation an better control of numerical oscillations near erivative singularities. Stability an convergence of the schemes are prove theoretically, an numerical simulation results are provie to compare with the scheme in []. AMS subject classification: 65M6, 37K Key wors: iscontinuous Galerkin metho, local iscontinuous Galerkin metho, issipation, Hunter-Saton equation, stability Department of Mathematics, University of Science an Technology of China, Hefei, Anhui 36, P.R. China. yu@ustc.eu.cn. Research supporte by NSFC grant 655, FANEDD of CAS an SRF for ROCS SEM. Division of Applie Mathematics, Brown University, Provience, RI 9, USA. shu@am.brown.eu. Research supporte by NSF grant DMS-8986 an ARO grant W9NF

2 Introuction In this paper, we present further evelopment of the local iscontinuous Galerkin (LDG) metho esigne in [] an a new issipative iscontinuous Galerkin (DG) metho for the Hunter-Saton (HS) equation u t + u u + uu =. (.) In [], we evelope a LDG metho for the HS type equations an gave a rigorous proof for its energy stability. In this metho the basis functions use are iscontinuous in space. The LDG iscretization also results in a high orer accurate, etremely local, element base iscretization. In particular, the LDG metho is well suite for hp-aaptation, which consists of local mesh refinement an/or the ajustment of the polynomial orer in iniviual elements. Numerical simulation shows that the LDG metho in [] approimates the issipative regularization an the ispersive regularization, as well as continuous solutions of the original HS equation (with possibly iscontinuous erivatives) quite well. However, when the erivative has a big jump iscontinuity, the LDG solution for the erivative contains spurious numerical oscillations, which are controlle by a nonlinear limiter in []. In this paper, we attempt to improve the performance of the LDG metho in its control on spurious numerical oscillations near erivative singularities, without sacrificing its accuracy an provable stability. We also esign a new issipative DG metho with the same improve numerical performance an provable convergence for the piecewise constant case. The DG metho is a class of finite element methos, using iscontinuous, piecewise polynomials as the solution an the test space. It was first esigne as a metho for solving hyperbolic conservation laws containing only first orer spatial erivatives, e.g. Ree an Hill [5] for solving linear equations, an Cockburn et al. [4, 3,, 6] for solving nonlinear equations. It is ifficult to apply the DG metho irectly to the equations with higher orer erivatives. The LDG metho is an etension of the DG metho aime at solving partial ifferential equations (PDEs) containing higher than first orer spatial erivatives. The first LDG metho was constructe by Cockburn an Shu in [5] for solving nonlinear convection iffusion equations containing secon orer

3 spatial erivatives. Their work was motivate by the successful numerical eperiments of Bassi an Rebay [] for the compressible Navier-Stokes equations. The iea of the LDG metho is to rewrite the equations with higher orer erivatives into a first orer system, then apply the DG metho on the system. The esign of the numerical flues is the key ingreient to ensure stability. The LDG techniques have been evelope for convection iffusion equations (containing secon erivatives) [5], nonlinear one-imensional an two-imensional KV type equations [, 9] an the Camassa-Holm equation []. More general information about DG methos for elliptic, parabolic an hyperbolic partial ifferential equations can be foun in the two special journal issues evote to the DG metho [7, 8], as well as in the recent books an lecture notes [4, 9, 6, 7]. This paper is organize as follows. In Section, we present an analyze our improve, issipative LDG metho for the HS type equations (.). We give a proof of the energy stability in Section.3. In Section 3, we present a new issipative DG metho for the HS equation. Stability for general case is prove, as well as convergence for the piecewise constant P case. Section 4 contains numerical results to compare with the results in [] an to emonstrate the accuracy an capability of the methos. Concluing remarks are given in Section 5. The issipative LDG metho for the HS equation. Notation We enote the mesh in [, L] by = [ j, j+ ], for j =,..., N. The center of the cell is j = ( j + j+ ) an the mesh size is enote by h j = j+ j, with h = ma j N h j being the maimum mesh size. We assume that the mesh is regular, namely that the ratio between the maimum an the minimum mesh sizes stays boune uring mesh refinements. We efine the piecewise-polynomial space V h as the space of polynomials of the egree up to k in each cell, i.e. V h = {v L (Ω) : v P k ( ) for, j =,..., N}. Note that functions in V h are allowe to have iscontinuities across element interfaces. 3

4 The solution of the numerical scheme is enote by u h, which belongs to the finite element space V h. We enote by (u h ) + an (u j+ h ) the values of u j+ h at j+, from the right cell +, an from the left cell, respectively. We use the usual notations [u h ] = u + h u h an ū h = (u+ h + u h ) to enote the jump an the mean of the function u h at each element bounary point, respectively.. The LDG metho In this section, we efine our LDG metho for the HS equations (.), written in the following form u = q, (.) q t = ((u ) ) (u ) (.) with an initial conition u(, ) = u (), (.3) an the bounary conitions u(, t) = u (, t) = u (L, t) =, (.4) or u(l, t) = u (, t) = u (L, t) =. (.5) To efine the local iscontinuous Galerkin metho, we further rewrite the equation (.) as a first orer system: q r =, (.6) r u =. The LDG metho for the equations (.6), where q is assume known an we woul want to solve for u, is formulate as follows: fin u h, r h V h such that, for all test functions ρ, ϑ V h, q h ρ + r h ρ ( r h ρ ) j+ 4 + ( r h ρ + ) j =, (.7a)

5 r h ϑ + u h ϑ (û h ϑ ) j+ + (û h ϑ + ) j =. (.7b) The hat terms in (.7) in the cell bounary terms from integration by parts are the so-calle numerical flues, which are single value functions efine on the eges an shoul be esigne base on ifferent guiing principles for ifferent PDEs to ensure stability. For the stanar elliptic equation (.6), we can take the simple choices such that r h = r + h, û h = u h, (.8) where we have omitte the half-integer inices j+ as all quantities in (.8) are compute at the same points (i.e. the interfaces between the cells). Equation (.) can be rewritten as a first orer system: q t + p B(r) =, p (b(r)u) =, (.9) r u =, where B(r) = r an b(r) = B (r) = r. Now we can efine a local iscontinuous Galerkin metho to equations (.9), resulting in the following scheme: fin q h, p h, r h V h such that, for all test functions ϕ, ψ, η V h, (q h ) t ϕ p h ψ + I j r h η + (p h B(r h ))ϕ + (( p h B(r h ))ϕ ) j+ b(r h )u h ψ ( b(r h )ũ h ψ ) j+ u h η (û h η ) j+ + (û h η + ) j The numerical flues in equations (.) are chosen as + ( b(r h )ũ h ψ + ) j (( p h B(r h ))ϕ + ) j =, (.a) =, (.b) =. (.c) p h = p + h, û h = u h, B(r h ) = B(r + h ), ũ h = u h (.) an r b(r h if u h h ) =, (.) r + h if u h <. 5

6 The bounary conitions for the LDG scheme of the HS equation are taken as (u h ) =, (r h ) =, (r h ) + N+ =, (p h ) + N+ =. (.3) The efinition of the algorithm is now complete. Notice that we have use the same notation r h in (.7b) an in (.c) as the schemes as well as the flues chosen are ientical. The implementation etail of the algorithm can be foun in []. Remark.. Comparing with the scheme in [], the main ifference of the scheme in this paper is the choice of the numerical flu for b(r h ). The numerical flu in [] is the central flu, i.e. b(r h ) = B(r+ h ) B(r h ) r + h r h = (r+ h + r h ). (.4) The issue of convergence to its weak solutions of the LDG metho in [] is subtle. It seems that the LDG scheme in [] without limiters, if converges, shoul converge to the conservative solution because the numerical solution is energy conservative. The numerical flu (.) is base on the upwining principle. The resulting scheme provie aitional energy issipation, thus facilitating convergence towars the issipative weak solution without any limiters. Remark.. We remark that the choice for the flues is not unique. We can also choose the following numerical flues r h = r h, û h = u + h, p h = p h, B(r h ) = B(r h ), ũ h = u + h, (.5) r b(r h if u + h h ) =, (.6) r + h if u + h <..3 Energy stability of the LDG metho In this section, we prove the energy stability of the LDG metho for the HS type equations efine in the previous section. 6

7 Proposition.. (Energy stability) The solution to the schemes (.7) with flues (.8) an (.) with flues (.) (.) satisfies the energy stability t L rh. (.7) Proof. For equation (.7a), we first take the time erivative an get (q h ) t ρ + (r h ) t ρ ( (r h ) t ρ ) j+ + ( (r h ) t ρ + ) j =. (.8) Since (.8), (.7b) an (.a) (.c) hol for any test functions in V h, we can choose ρ = u h, ϑ = (r h ) t, ϕ = u h, ψ = r h, η = p h. With these choices of test functions an summing up the five equations in (.8), (.7b) an (.a) (.c), we obtain (r h ) t r h + (p h u h ) ( p h u h + û hp h ) j+ + ( p h u + h + û hp + h ) j I j (B(r h )u h ) + ( B(r h )u h + b(r h )ũ h r h ) j+ ( B(r h )u + h + b(r h )ũ h r + h ) j I j + ((r h ) t u h ) ( (r h ) t u h + û h(r h ) t) j+ + ( (r h ) t u + h + û h(r + h ) t) j =. We have (r h ) t r h + Ψ j+ where the numerical entropy flues are given by Ψ j+ Ψ j + Θ j =, (.9) ( = p h u h ( p hu h + û hp h ) B(r h )u h + B(r h )u h + b(r ) h )ũ h r h + (r h ) tu h ( (r h ) t u h + û h(r h ) t) an the etra term Θ is given by Θ j ( = [p h u h ] + p h [u h ] + û h [p h ] + [B(r h )u h ] b(r h )ũ h [r h ] B(r h )[u h ] [(r h ) t u h ] + (r ) h ) t [u h ] + û h [(r h ) t ] With the efinition (.8) an (.) of the numerical flues an after some algebraic manipulation, we easily obtain [p h u h ] + p h [u h ] + û h [p h ] =, 7 j+, j.

8 [(r h ) t u h ] + (r h ) t [u h ] + û h [(r h ) t ] =, an Hence [B(r h )u h ] b(r h )ũ h [r h ] B(r h )[u h ] (.) = ((r+ h ) B(r h ))(u + h u h ) + (r+ h r h ) ( (r+ h + r h )u h b(r h )ũ h ) (.8) ( = (r + h r h ) (r+ h + r h )u h b(r h )u h (r+ h r h ) u h if u h, (.5) =. (r+ h r h ) u h if u h <. Θ j ). (.) Summing up the cell entropy inequalities ((.9) with (.)) an taking care of the bounary conitions (.3), we obtain (r h ) t r h. (.) This is the esire energy stability (.7). I Remark.3. The proof is similar to that in []. However, the numerical flues (.5) provie aitional energy issipation in (.). Remark.4. For the numerical flues (.5) an (.6), we can also prove the energy stability (.7). 3 A new issipative DG metho for the HS equation In [], the HS equation is also written in the following form (u t + uu ) = (u ) (3.) with u(, t) =, t [, T ] (3.) 8

9 an u(, ) = u (). (3.3) By introucing r = u, (3.4) we may write the equation as r t + (ur) = r, r = u (3.5) or r t + ur = r, r = u. (3.6) In [], Holen et al. prove that a upwin finite ifference metho approimating (3.6) converges to the unique issipative solution of the HS equation (3.6). The issipative solutions for the HS equation are well stuie from a mathematical point of view, we refer the reaers to a series of papers [, 3, 3, 4, 5]. This motivates us to consier a new issipative DG metho which, as we will see later, becomes equivalent to the first orer upwin finite ifference metho in [] for the piecewise constant P case, thus convergent to the unique issipative solution of (3.6) in this case. Therefore, this issipative DG metho can be consiere as a irect generalization of the metho in [] to high orer accuracy. In the following, we will present this new issipative DG metho an prove its stability. We will also emonstrate that this DG scheme, for the piecewise constant P case, is equivalent to the upwin finite ifference scheme in []. 3. The new DG metho The new DG scheme for the equation (3.5) is efine as follows. Fin r h V h such that, for all test functions ρ V h, (r h ) t ρ u h r h ρ + (u h r h ρ ) j+ 9 (u h r h ρ + ) j = r h ρ, (3.7)

10 where u h = r h (, t), (3.8) which is a continuous piecewise polynomial function with egree at most k +. The numerical flu r h in (3.7) is taking as r h if u h, r h = r + h if u h <. (3.9) which is also base on the upwining principle. The bounary conitions for the DG scheme of the HS equation are taken as (u h ) =, (r h ) =, (r h ) + N+ =. (3.) The efinition of the algorithm is now complete. Remark 3.. The main ifference between the DG scheme (3.7) an the LDG scheme (.7) an (.) in Section is the solution u h. u h in (3.7) is a continuous piecewise polynomial function of egree k +. u h in (.7) an (.) in Section is a iscontinuous piecewise polynomial function of egree k. The approimations to u in both schemes are iscontinuous piecewise polynomial functions of egree k. 3. Stability of the DG metho Proposition 3.. (Stability) The solution to the scheme (3.7) (3.9) satisfies the stability t L rh. (3.) Proof. Choosing the test function ρ = r h in (3.7), we obtain (r h ) t r h u h r h (r h ) + (u h r h r h ) j+ (u h r h r + h ) j = r 3 h.

11 Using integration by parts for the secon term in the above equation an noticing the continuity of u h, we have (r h ) t r h + (u h ) rh (u h(r h ) ) j+ + (u h r h r h ) j+ (u h r h r + h ) j = r 3 h. + (u h(r + h ) ) j (3.) The secon term an the last term in the equation (3.) will be cancelle because (u h ) = r h. Now the equation (3.) can be written in the following form (r h ) t r h + Φ j+ Φ j + Θ j =, (3.3) where the numerical entropy flues are given by an the etra term Θ is given by Φ = u h r h r h (u h(r h ) ) Θ = u h r h r h u h r h r + h (u h(r h ) ) + (u h(r + h ) ). With the efinition (3.9) of the numerical flues an after some algebraic manipulation, we easily obtain Θ = u h (r + h r h )( (r+ h r h ) r h) = (r+ h r h ) u h if u h, (r+ h r h ) u h if u h <.. Hence Θ. (3.4) Summing up the cell entropy inequalities ((3.3) with (3.4)) an taking care of the bounary conitions (3.), we obtain (r h ) t r h. (3.5) This is the esire stability result (3.). I

12 3.3 Convergence of the DG metho for the P case In this section, we iscuss the convergence of the DG metho in Section 3. for the piecewise constant approimation, namely, iscontinuous P finite elements. We enote by v j, the numerical solution v h in the cell (it is a constant in each cell). Then from (3.8), we can get (u h ) j (u h ) j+ j = (v h ) i h i (3.6) = i= j (v h ) i h i, (3.7) i= (u h ) j+ (u h ) j The scheme (3.7) (3.9) can be rewritten in the eplicit form = (v h ) j h j. (3.8) t (v h) j h j + (u h ) j+ (v h ) j (u h ) j (v h ) j = (v h) jh j, if u h >, (3.9a) t (v h) j h j + (u h ) j+ (v h ) j+ (u h ) j (v h ) j = (v h) j h j, if u h <. (3.9b) In the following, we will prove that the scheme (3.9) can be written in the following form t (v h) j + (u h ) j D (v h ) j = (v h) j, if u h >, (3.a) t (v h) j + (u h ) j+ D + (v h ) j = (v h) j, if u h < (3.b) where D (v h ) j = ( (v h ) j (v h ) j ) /hj an D + (v h ) j = ( (v h ) j+ (v h ) j ) /hj. This is just the semi-iscrete finite ifference scheme in []. Lemma 3.. The scheme (3.9) is equivalent to the scheme (3.). Proof. We will prove the Lemma for u h > an u h < respectively. u h >. The scheme (3.9a) can be written as t (v ( ) ( h) j h j + (u h ) j (vh ) j (v h ) j + (uh ) j+ ) (u h ) j (vh ) j = (v h) j h j.

13 Now, using the relation in (3.8), we can obtain That is (3.a) u h <. t (v ( ) h) j h j + (u h ) j (vh ) j (v h ) j + (vh ) j h j = (v h) j h j. The scheme (3.9b) can be written as t (v h) j + (u h ) j D (v h ) j = (v h) j. t (v ( ) ( h) j h j + (u h ) j+ (vh ) j+ (v h ) j + (uh ) j+ Now, using the relation in (3.8), we can obtain That is (3.b) ) (u h ) j (vh ) j = (v h) j h j. t (v ( ) h) j h j + (u h ) j+ (vh ) j+ (v h ) j + (vh ) jh j = (v h) jh j. t (v h) j + (u h ) j+ D + (v h ) j = (v h) j. Remark 3.. The equivalence of the schemes (3.9) an (3.) implies that we can irectly get the convergence results for P DG metho since the convergence results were prove for (3.) in []. The numerical solution with P polynomial functions will converge to a issipative solution of HS equation. Remark 3.3. Because of the equivalency in the P case, the issipative DG metho can be consiere as a irect generalization of the metho in [] to high orer accuracy. The convergence proof for the DG scheme in P k with k is however more complicate an is left for future work. 4 Numerical results We will first show the accuracy test results for the upwining numerical schemes in this paper. We also present one numerical eample (Eample 4. in []) here, which contains 3

14 the solution with iscontinuous erivative, to compare ifferent methos an to show the avantage of the schemes in this paper. The results for other eamples in [] are similar for all the schemes an hence are not shown to save space. Time iscretization is by the thir orer eplicit TVD Runge-Kutta metho in [8]. This is not the most efficient metho for the time iscretization to our LDG scheme. However, we will not aress the issue of time iscretization efficiency in this paper. Consier the numerical solution of the HS equation u t + u u + uu =. (4.) First, we test our metho taking the eact solution u(, t) =.( π) ( + π) sin( t) (4.) for the HS equation with a source term f, which is a given function so that (4.) is the eact solution. The computational omain is [ π, π]. The L errors for the numerical solutions of u h, r h an the numerical orers of accuracy at time t =.5 with uniform meshes are containe in Table 4.. We can see that our schemes in this paper have high orer accuracy for the smooth solutions. The observation for the accuracy orer is the following: The LDG metho in Section with the upwining numerical flu (.). The metho with P k elements gives a uniform (k+)-th orer of accuracy for u h. For the solution r h, the accuracy is k-th orer for k. The DG metho in Section 3 with the upwining numerical flu (3.9). The metho with P k elements gives a uniform (k+)-th orer of accuracy for u h. For the solution r h, the accuracy is (k+)-th orer for k. From Table 4., we can see that, for the same mesh, the magnitue an orer of the error for the DG metho are better than those for the LDG metho. This is consistent with the ifferent choices of the polynomial egree of u h for the two methos. 4

15 Table 4.: L errors an orers of accuracy for the LDG metho in Section with the upwining numerical flu (.) an the DG metho in Section 3 with the upwining numerical flu (3.9) for the eact solution (4.), t =.5. LDG, upwining flu DG, upwining flu u h r h u h r h N L error orer L error orer L error orer L error orer 3.6E- 8.7E-.84E- 4.53E- P.76E E E-..76E E-..5E E-.5.45E E-3..6E-.99.6E E E-.77E- 9.E-3.E- P.93E E-.95.5E E E E-..33E E E E-.5.6E E E E- 7.97E-4.43E- P.8E E E E E E E E E E E E E-4.9E-.94E-4 5.8E-3 P 3 5.6E E E E E E E E E E-5.96.E E Net, we consier the initial conition, if, u(, ) =, if < <,, if. (4.3) 5

16 The eact solution of the HS equation (4.) with the initial conition (4.3) is, if, u(, t) =, if < < (.5t + (.5t+) ), (.5t + ), if (.5t + ). (4.4) The computational omain is [ 6, 6]. Even though the solution u is continuous, the lack of smoothness of u will introuce high-frequency oscillation into the calculation of the resiual. To compare the performance of the ifferent methos, we use the following methos in the computations by P elements with N = 8 cells for the solution u an r (i.e. u ). (a) The LDG metho in [] with the central numerical flu (.4). (b) The LDG metho in [] with the central numerical flu (.4) an the total variation boune in the means (TVBM) limiter in [4] to control the oscillation of r h. (c) The LDG metho in Section with the upwining numerical flu (.). () The DG metho in Section 3 with the upwining numerical flu (3.9). We show the numerical solutions u h in Figures an r h in Figures We can see that the central scheme (a) without limiter generates spurious oscillation in r h approimating u near its iscontinuity, especially at the later time t =.5. The TVBM limiter (b) removes these oscillations in r h, at the price of smearing the erivative iscontinuity an smoothing out the solution u as well. The numerical solutions base on the upwining schemes (c) () have much sharper fronts than those with the TVBM limiter, with just slight, localize over an uner-shoots near the fronts. In Figure 4.5, the energy L r h as a function of time for the numerical solutions is shown. The solution of r for scheme (a) is oscillatory without the limiter, but the energy is still conserve. When the limiter is use, the energy is ecaying a lot. The upwining schemes (c) () provie aitional energy issipation (but not too much), thus facilitating convergence towars the weak solution without any limiters. 6

17 .4 t=..4 t= u.6 u (a) LDG, central flu (b) LDG, limiter.4 t=..4 t= u.6 u (c) LDG, upwining flu () DG, upwining flu Figure 4.: Solution u for the HS equation (4.) the initial conition (4.3) for ifferent schemes at t =.. 5 Conclusion We have evelope issipative LDG an DG methos to solve the HS type equations. Energy stability is prove for general solutions for both schemes. Convergence of the DG scheme for the P case is prove by showing its equivalency with the upwin scheme in []. An important issue not aresse in this paper is convergence proof for high orer DG methos. From the stability an approimation results, we can erive L a priori 7

18 .4 t=.5.4 t= u.6 u (a) LDG, central flu (b) LDG, limiter.4 t=.5.4 t= u.6 u (c) LDG, upwining flu () DG, upwining flu Figure 4.: Solution u for the HS equation (4.) the initial conition (4.3) for ifferent schemes at t =.5. error estimates of the high orer LDG metho for the Camassa-Holm equation []. However the proof of the high orer DG metho for the HS equation is not completely straightforwar. Such proof is left for future work. 8

19 .4 t=..4 t= r.6 r (a) LDG, central flu (b) LDG, limiter.4 t=..4 t= r.6 r (c) LDG, upwining flu () DG, upwining flu Figure 4.3: Solution r (i.e. u ) for the HS equation (4.) the initial conition (4.3) for ifferent schemes at t =.. References [] F. Bassi an S. Rebay. A high-orer accurate iscontinuous finite element metho for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys., 3:67 79, 997. [] B. Cockburn, S. Hou, an C.-W. Shu. The Runge-Kutta local projection iscontinuous Galerkin finite element metho for conservation laws IV: the multiimensional 9

20 .4 t=.5.4 t= r.6 r (a) LDG, central flu (b) LDG, limiter.4 t=.5.4 t= r.6 r (c) LDG, upwining flu () DG, upwining flu Figure 4.4: Solution r (i.e. u ) for the HS equation (4.) the initial conition (4.3) for ifferent schemes at t =.5. case. Math. Comp., 54:545 58, 99. [3] B. Cockburn, S.-Y. Lin, an C.-W. Shu. TVB Runge-Kutta local projection iscontinuous Galerkin finite element metho for conservation laws III: one imensional systems. J. Comput. Phys., 84:9 3, 989. [4] B. Cockburn an C.-W. Shu. TVB Runge-Kutta local projection iscontinuous Galerkin finite element metho for conservation laws II: general framework. Math.

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