1. Introduction. In this paper we consider the nonlinear Hamilton Jacobi equations. u (x, 0) = ϕ (x) inω, ϕ L (Ω),

Transcription

1 SIAM J. UMER. AAL. Vol. 38, o. 5, pp c 000 Society for Inustrial an Applie Mathematics SPECTRAL VISCOSITY APPROXIMATIOS TO HAMILTO JACOBI SOLUTIOS OLGA LEPSKY Abstract. The spectral viscosity approximate solution of convex Hamilton Jacobi equations with perioic bounary conitions is stuie. It is prove in this paper that the approximation an its graient remain uniformly boune, formally spectral accurate, an converge to the unique viscosity solution. The L 1 -convergence rate of the orer 1 ε ε >0 is obtaine. Key wors. Hamilton Jacobi equation, viscosity solution, spectral viscosity metho, vanishing viscosity metho, convergence rate, error estimate AMS subject classifications. 35L65, 65M70, 65M15 PII. S Introuction. In this paper we consier the nonlinear Hamilton Jacobi equations 1 t u x, t+h x, t, x u x, t = 0 in Ω [0,T], 0 <T <, u x, 0 = ϕ x inω, ϕ L Ω, with Hamiltonian H x, t, p strictly convex with respect to x an p: Hxx H αi xp βi 0 <α,β<, H px H pp where I enotes a unit matrix. We also assume that Ω = [0, π], = 1,, 3,..., the initial conition ϕ x, an the Hamiltonian H x, t, p are π-perioic in x. Equations of Hamilton Jacobi type arise in many areas of application, incluing the calculus of variations, control theory, ifferential games, image processing see, e.g., [17], [1]. The generalize solutions to 1 are Lipschitz continuous hence ifferentiable almost everywhere, but may have iscontinuous erivatives, regarless of the smoothness of the initial conition ϕ x [17]. Solutions with such iscontinuities are not unique. The efinition of the viscosity solution to 1, its well-poseness in L an other properties were formulate an systematically stuie by Kruzhkov, Lions, Cranall, Evans, Souganiis, an others [1], [17], [6], [8], [3]. Following these results, first an higher orer numerical methos were evelope for Hamilton Jacobi equations: finite ifference methos [7], [3], [0], [10], [15], [16], finite volume methos [1], [13], finite element methos [3], [9], [14]. Recently, Lin an Tamor provie the convergence framework for general approximate solutions of multiimensional Hamilton Jacobi equations [15], which we use in the paper. In this paper we suggest the numerical solution of the π-perioic initial value problem 1 in R 1 by a spectral viscosity metho. We prove see Theorem 5. that the numerical solution converges to the exact unique viscosity solution of 1 an obtain the L 1 -convergence rate of the orer 1 ε ε > 0. The spectral Receive by the eitors March 10, 1999; accepte for publication in revise form June 13, 000; publishe electronically ovember 17, Department of Mathematics, Brown University, Provience, RI 0906 olga@math.brown.eu. 1439

2 1440 OLGA LEPSKY viscosity metho was introuce an analyze for scalar conservation laws an certain hyperbolic systems of conservation laws by Tamor an his coworkers see, e.g., [4], [], [5], [5], [19], [18]. In the one-imensional case the erivative of the viscosity solution of convex Hamilton Jacobi equations 1 is equivalent to the entropy solution of scalar conservation laws. However, in the multiimensional case the graient of the viscosity solution of 1 satisfies a weakly hyperbolic system of conservation laws [1], [17], [11], for which the convergence of the spectral viscosity metho was not proven. The following abbreviations are use throughout this paper: t = t, s j = s x s, jk s = s j x s, j xs x s = 1, s,..., s s, k x = x, Dx is a matrix with entries jk. To solve the 1 by a spectral viscosity metho we approximate the spectral pseuospectral projection of the viscosity solution P u by an -trigonometric polynomial 3 u = û ξ te iξ x, ξ i which is governe by the semiiscrete approximation 4 t u x, t+p H x, t, x u x, t = ε j Q j x, t u x, t. Together with one s favorite ODE solver, 4 gives a fully iscrete metho for the approximate solutions of 1. Following [5], we use the spectral viscosity ε j Qj x, t u x, t. We show that the spectral viscosity is small enough to retain the formal spectral accuracy of the overall approximation, while sufficiently large to enforce the uniform stability an L 1 -convergence of the approximation u x, t to the unique viscosity solution see Lemma 3.1, Lemma 3., an Theorem 5.. It consists of the following three ingreients: i a vanishing viscosity amplitue ε, 5 ε θ,θ 0, 1; ii a viscosity-free spectrum of size m 1. For the proof of W 1, -stability of u an convergence of the truncation error to zero, it is enough to take m δ, 0 <δ<θ,while for the proof of semiconcave stability, an hence, convergence we use m θ/4 6 log ; 1/4 iii a family of viscosity kernels Q j x, t, activate only on high wave numbers ξ = 1/ i=1 ξ i m, such that 7 ε Q j x, t = j Q j x, t u x, t = ε Qj teiξ x ξ, ξ i, ξ m ξ i, ξ m Q j ξ tξ j û ξ te iξ x.

3 SPECTRAL VISCOSITY METHOD FOR HAMILTO JACOBI 1441 Here we take the Fourier coefficients Q j ξ t such that Q j ξ t = Q j pt, when ξ = i=1 ξ i 1/ = p, 8 Q j pt =0 p <m, 1 Const m4 p 4 Q j pt 1 p m, Q j pt Q j p+1 t. Define also 9 j R j x, t = j Q j x, t. Then we may rewrite the semiiscrete approximation 4 as 10 t u x, t+h x, t, x u x, t ε u x, t =I P H x, t, x u x, t ε j R j x, t u x, t. The convergence analysis of the spectral viscosity metho is base on its close resemblance to the vanishing viscosity approximation 11 t u ε x, t+h x, t, x u ε x, t ε u ε x, t =0. Here ux, t = lim ε 0 u ε x, t inw 1, is the unique viscosity solution. The paper is organize as follows. In section we obtain the L p -estimates 1 p of the terms on the right of 10. These estimates enable us to show in section 3 that the spectral viscosity approximation u x, t an its graient remain uniformly boune with the growth of. In section 4 we prove the semiconcave stability of the approximate solution. Combining this result with the ecay rate of the truncation error, obtaine in section 5, an applying Lin Tamor convergence theorem [15], we conclue the convergence of the metho an obtain its L 1 -convergence rate Theorem 5... Preliminary estimates. In this section we will estimate the two terms on the right of 10. In view of Theorem 9.1of [6] p 1we have ε j R j u 1 ε j R j u Lp x L p, x L 1 x ε j R j u 13 ε j R j u L 1 x. L p x Lemma.1. ε j s R j x, t L C 0 ε m s 1 log x L p x C 1 log θ, s =0, C 1 log θ 3/4, s =1,

4 144 OLGA LEPSKY where s {0, 1} ; C 0,C 1 o not epen on. Proof. This lemma correspons to Lemma 3.1in [5]. Lemma.. ε j s R j x, t L x C 0 ε m s+ log { 1/ C log 1, θ s =0, C 1, s =1, where s {0, 1} an C 0,C 1 o not epen on. Remark. Lemma. is a multiimensional generalization of Lemma 3.1in [5]. Proof. By 9 an 8 for q = an ξ = p { 1if p<m, R pt j = 1 Q j m q pt Const p if p m. Then for s 0 ε j s R j x, t L x = ε j R pt j e iξ x ε s s j p=0 m 1 p=0 +Const ε ξ i, ξ =p e iξ x ξ i, ξ =p L x s j p=m L x m q p ξ i, ξ =p m s+ e iξ x L x m 1 Const ε p s e iξ x p=0 ξ i, ξ =p L x q + Const ε m p q s e iξ x p=m ξ i, ξ =p L x m 1 q Const ε p s+1 + m p=0 L x p q s 1 p=m L x { } m s+ Const ε m s+ if q s 1 > 1, + log if q s 1=1 Const ε m s+ log, since q = 1+s. Here is the space imension. ow, using 5 an 6 completes the proof of the lemma. ote. For the simplicity of the proofs from now on we will assume that Hamiltonian epens only on u, i.e., H = H u, such that 14 αi H pp p βi 0 <α,β<, p =p 1,...,p,Iis a unit matrix.

5 SPECTRAL VISCOSITY METHOD FOR HAMILTO JACOBI 1443 However, all the results an proofs of this paper are easily extene to the case H = H x, t, u satisfying. Lemma.3. Denote Then s 1 15 p α H p := p α1 1 α p H p, H Ck x : = max p α H p, α =k L B { } B := p : p x u L x. xh s x u x, t L x K s x s+1 u L, x where K s epens on x u L x, H, an s : 16 K s s r=1 H C r x xu r 1 L x. Proof. Define a multi-inex α =α 1,...,α r, α i {1,,...,s}, α = r l=1 α l. By the chain rule we have s j s r H α H x u x, t = 1 j i1 u α r j ir u. p r=1 α =s i 1=1 i i1 p ir r=1 17 The Höler inequality followe by the Gagliaro irenberg inequality implies that 18 α1 j r k=1 C 0 C 0 i1 u α r j ir u L x α k j ik u L s/α k x r m=1 k=1 m=1 s j m u α k /s r m=1 k=1 α k j L x mu 1 α k/s L x s j m u L x mu r 1 L x. m u L s/α k x By Parseval s ientity followe by the Young inequality j s m u L = 1/ 19 ξ s x j ξm û ξ ξj s ξm û ξ ξ ξ j ξm û ξ ξ s 1/ ξ ξ 1/ s +1 ξ s+1 ûξ ξ s +1 s+1 x u L x. ξ s+1 j s+1 s + ξs+1 m 1 s+1 1/ û ξ 1/

6 1444 OLGA LEPSKY Combining 17, 18, an 19 we obtain xh s x u x, t L x C s x s+1 u L H x C r x u r 1 L x. r=1 3. Uniform bouneness of the approximate solution an its graient. We begin with the following facts: i In view of the approximation error estimate see section 9.7 of [4] s r 0 there hols 0 x r I P H x u x, t L x C s r s xh x u x, t L x. ii The Sobolev imbeing theorem see, e.g., [] combine with i implies 1 x r I P H x u x, t L x I P H x u x, t x r+1+[/] L x C s r 1 [/] s xh x u x, t L x. Here [/] = max {z Z, z /}; C is a constant inepenent of. Assumptions. From now on we assume that our ata the initial conitions u x, 0 an the flux H x ux, t satisfy the following conitions: a xu s x, 0 L Const for s =0, 1; b xu s x, 0 L Const/ε s 1/ for s ; c The flux H is sufficiently smooth; that is, x u L x,t < implies H x u C s x, t < for sufficiently large s. Lemma 3.1. Consier the spectral viscosity approximation 4. Then for any s hols 3 4 s xu L x,t C xu ε s 1 s xu L x C xu ε s 1/ s xu L x C xu ε s s p=1 s 1 p=1 s 1 p=1 K p L t +1, K p L t +1, K p L t +1. Here C x u = H x u x, t C 0x + xu L x +1, K s is efine by 16 for s 1, s = s +1/+[/].

7 SPECTRAL VISCOSITY METHOD FOR HAMILTO JACOBI 1445 Proof. Multiplying 10 by s x u x, t,s 1an integrating in x yiels 5 1 t s xu L x + ε x s+1 u L x x s+1 u L s 1 x x P H x u x, t L x + xu s L x ε j R j x, t s xu L x. Using 1 an Lemma. for the first term on the right of 5 an the Cauchy Schwarz inequality for the secon term, we en up with 6 1 t s xu L x + ε x s+1 u L x C 1 xu s L x + 1 s 1 x P H x u x, t ε. L x It remains to estimate the last term. In view of the interpolation error estimate see section 9 of [4] an 15 s 1 x P H x u x, t L x s 1 x H x u x, t + L x s 1 x I P H x u x, t L x C s 1 x H x u x, t L x { Ks 1 s C xu L x for s, K 0 for s =1. Substituting this estimate to 6 yiels t xu L x + ε x u C L x 1 x u L x + C K0, ε t s xu L x + ε The temporal integration implies s+1 x u L x C 1 + C K s 1 ε x u L x + ε xu L x,t s xu L x for s. x u x, 0 L +C 1 x u L x,t + C K 0 L t ; ε xu s x, t L x + ε x s+1 u L x,t xu s x, 0 L + C 1 + C K s 1 L t ε xu s L x,t, an eliminating the squares gives

8 1446 OLGA LEPSKY x u L x + ε1/ xu L x,t x u x, 0 L + C 1 x u L x,t + K 0 L t C, ε 1/ xu s x, t L x + ε1/ x s+1 u L x,t C1 C xu s x, 0 L x + K s 1 L + t xu s ε 1/ L x,t. It follows that xu L x,t Const K0 L t ε + 1 ε 1/ s+1 x u L x,t Const Ks 1 L t ε + 1 s xu x, t L x Const Ks 1 L t ε 1/ x u L x,t + 1 ε 1/ ε 1/, xu s L x,t sxu Lx,t +1, which in turn yiels the an 3 for ε sufficiently small. We apply the Sobolev inequality xu s L x x s+1+[/] u L x to obtain 4. Lemma 3. uniform bouneness of the approximation an its graient. Consier the spectral viscosity approximation 4. Then we have u x, t L x + xu x, t L x C t [0,T] for T<, where C oes not epen on. Proof. Differentiating 10 in x we obtain ε 1/ 7 t x u x, t+ x H x u x, t ε x u x, t = x I P H x u x, t ε j R j x, t xu x, t. otice that 8 1 xu x, t L x max j,x ju x, t = k u x 0,t. First assume that k u x 0,t > 0 t [0,T]. Then k H x u x 0,t = p j H p j k u = 0 an ε k u x 0,t 0. Denote v t := k u x 0,t; here k an x 0 epen on t. It is easy to show that v t is a Lipschitz continuous function. Therefore v t is ifferentiable almost everywhere a.e. by Raemacher s theorem. It follows from 7 that for a.e. t v t t x I P H x u x, t L x + ε x j R j x, t u x, t L x.

9 SPECTRAL VISCOSITY METHOD FOR HAMILTO JACOBI 1447 We use 1, 15, an 3 to estimate the first term on the right: x I P H x u x, t L x K s s+1 s [/] x u L x C 0 K s [/] θs+1/ 0 L t + xu L x +1 s K s p=1 K p L t +1 ; while 1 combine with Lemma.1 provies the estimate for the secon term: ε x j R j x, t u 3/4 x, t log C θ x u. L x Finally, applying 8 we obtain for a.e. t L x 9 v t C s t K s [/] θs+1/ s K p L t +1 p=0 + CK s s p=1 K p L t +1 3/4 log + C v t. s [/] θs+1/ θ Take s +[/]+5θ/4 1 θ ; then we have s [/] θ s +1/ 3θ/4 > 0. ow fix ε>0 an assume that t T such that 0 : 30 ε/ <v t v 0 ε, 0 v t v 0 ε/ t t. Then integrating 9 in time an applying the efinition of K s in the statement of Lemma 3.1, we get 31 v t v 0 C 0 ε, s log + C s [/] θs+1/ 1 ε θ 3/4, where C 0 ε, s an C 1 ε epen only on ε, s, an T. It follows that v t v 0 ε/ for big, which contraicts the assumption 30. From this contraiction an from the continuity of v t it follows that for any C>0 there exist 0 such that v t v 0 C 0 an t [0,T]. Since v t = x u x, t L x, an v t is continuous on [0,T], we obtain that in the consiere case 3 x u x, t L x xu x, 0 L + C.

10 1448 OLGA LEPSKY ext, assume that for k an x 0 satisfying 8 k u x 0,t < 0 t [0,T]. Define w t := k u x 0,t = min x k u x, t. Then k H x u x 0,t = pj H p j k u =0, ε k u x 0,t 0, w t is a Lipschitz continuous function, an we obtain from 7 t w t x I P H x u x, t L x ε x j R j x, t u x, t L x for a.e. t. Applying the same estimates as in the case k u x 0,t > 0 we prove 3 for the case when k u x 0,t < 0 t [0,T]. Finally, the case when k u x 0,t, satisfying 8, changes sign on [0,T] may be reuce to the previous two cases. ow we will use the same strategy to prove the uniform bouneness of u x, t. If u x, t L x = max x u x, t t [0,T], then w t := max x u x, t is a Lipschitz continuous function, an 10 yiels for a.e. t t wk t+h 0 I P H x u x, t L x ε j R j x, t u x, t L x Applying 1, 15, 3, an 1, Lemma.1 for the estimation of the first an secon term, respectively, an using the uniform bouneness of the graient we obtain for a.e. t 3/4 t w C s log t H C s 1 [/] θs+1/ θ w t. It follows that t w C s t H Cw t. s 1 [/] θs+1/ By the Gronwall lemma the last inequality yiels 33 e tc 1 0 w t C H 0 + C s + w s 1 [/] θs+1/ 0 e tc.

11 SPECTRAL VISCOSITY METHOD FOR HAMILTO JACOBI 1449 t [0,T]. ext, if u x, t L x min u x, t t [0,T], x then let w t :=min x u x, t an the similar arguments imply 1 e tc C s 34 0 w t H 0 + C s 1 [/] θs+1/ +w 0 e tc. Combining 33 an 34 we conclue that e tc u x, t L x 1 C H u x, 0 L e tc. Corollary 3.3. For any s an for any hols xu s L x = C s θs 1/, xu s L x,t = C s θs 1, xu s L x = C s θs+1/+[/]. C s s 1 [/] θs+1/ Proof. The corollary follows from Lemma 3.1combine with Lemma 3.. Corollary 3.4. For any s an for any hols 35 x r I P H x u x, t L x C s ps, p s = s r θ s +1/, x r I P H x u x, t L x,t C s γs, γ s = s r θs, x r I P H x u x, t L x C s qs, q s = p s 1 [/]. Proof. The corollary follows from Lemma 3., Corollary 3.3, an estimates 15, 0, an Semiconcavestability of theapproximatesolution. In orer to prove the main result of this section we nee the following auxiliary lemma. Lemma 4.1. Let w x, t = ξ u x, t = ξ,dxu x, tξ, where u x, t is a trigonometric polynomial, π-perioic in x. Then w L 1 x = w +. L 1 x Proof. w x, t w x, t + w x, t. By the Green theorem Ω w x, t x epens solely on the bounary ata which in our case of a perioic Cauchy problem vanishes. Therefore Ω w x, t x = Ω w x, t + x. Lemma 4. semiconcave stability of the approximation. The spectral viscosity approximation u x, t is semiconcave stable, i.e., kt L 1 0,T, T <, such that sup x Ω, ξ =1 ξ,d x u x, tξ + kt for t [0,T] an. Proof. Let w := ξ u x, t = ξ,dxu x, tξ. Differentiating 10 with respect to x j,x k, an taking the inner prouct with a constant unit vector ξ yiels 36 t w x, t+ ξ,dxu x, t DpH Dxu x, tξ + p H, x w = ε x w ε j R j x, t w + ξ,dx I P H x u x, t ξ.

12 1450 OLGA LEPSKY The strict convexity of H 14 an the Cauchy Schwarz inequality imply 37 ξ,d x u x, t D ph D xu x, tξ α D xu x, tξ αw. Let W t =w x 0,t = max x Ω w x, t. It is a Lipschitz continuous an therefore a.e. ifferentiable. Without loss of generality assume that W t > 0 for t [0,T]. Then x w x 0,t = 0, x w x 0,t 0, an by 36 an 37 we have for a.e. t t W + αw ε j R j w x 0,t+ ξ,dx I P H x u x 0,t ξ. Equation 13, Lemma., an Lemma 4.1 yiel I = ε j R j w x 0,t ε j R j w L x ε j R j w L 1 x =C 1 w + Ω C L 1 x 1W. By 35 L x II = ξ,d x I P H x u x 0,t ξ Thus, for a.e. t D x I P H x u x, t L x C s 3+[/] + θ/ Const, for s. s 3 [/] θs+1/ 1 θ t W + αw I + II C W + C 3,α>0. Integration in time of the last inequality yiels W t k t =W 0 e Ct + C 3 e C t 1, C k t L 1 0,T for T<. Lemma 4.1an Lemma 4. imply the following corollary. Corollary 4.3. For any vector ξ =1, ξ,dxu x, tξ L Ω Ckt. 1 x 5. Convergence of the metho an the error estimate. To prove convergence we get the following truncation error estimate. Lemma 5.1. Define the truncation error F u = t u + H x u. Then hols 38 F u L 1 x,t = C θ log + C s s+sθ.

13 SPECTRAL VISCOSITY METHOD FOR HAMILTO JACOBI 1451 Moreover, if s u L x,t Const for some s an 0, then 39 F u L 1 x,t C s s/4+1/θ log + C s s+1. Proof. From 4 it follows that F u L 1 x,t ε j Q j x, t u x, t L 1 x,t In view of 9, Corollary 4.3, 1, an Lemma.1 we have I = ε j Q j x, t u x, t L 1 x,t = u x, t ε R j x, t j u x, t + I P H x u x, t L 1 x,t. L1 x,t ε u x, t L 1 x,t + ε R j x, t L1 j u x, t x,t L1 x,t Cε + C θ log C θ log. By II = I P H x u x, t L 1 x,t C s s+sθ s. It follows that II Const θ s θ 1 θ. ow 38 follows from estimates for I an II. ext, assume s u L x,t Const s. It follows from 7, 6, an the approximation error estimate [4] that for s I = ε j Q j x, t u x, t L 1 x,t C sθ/4 ε x s j Q j x, t u x, t L x,t = C sθ/4 ε j Q j x, t s xu x, t. Using 9, 1, an Lemma.1 we obtain I C sθ/4 L x,t ε + ε R j x, t L 1 j xu s x, t x,t L x,t C 1+s/4θ log j s xu x, t L x,t.

14 145 OLGA LEPSKY Finally, by Parseval s ientity an Young s inequality hencefore, j xu s x, t C L x,t s x s+ u x, t ; L x,t I C s 1+s/4θ log s+ x u x, t L x,t. In orer to estimate II we use 0, 15, an 16 : II = I P H x u x, t L 1 x,t C s s+1 s+ x u x, t L x,t. Since the two last inequalities hol for any integer s 0, the secon estimate of the lemma 39 follows. The proof of the lemma is complete. Theorem 5. convergence of the spectral viscosity metho. Let ux, t be the unique viscosity solution of the π-perioic initial value problem 1 an let u x, t be the spectral viscosity approximation, i.e., -trigonometric polynomial satisfying 4. Then for t [0,T] an we have the boun ux, t u x, t L 1 x C T ux, 0 u x, 0 L 1 x + C T θ log + Cs,T s1 θ. Moreover, if s u L x,t Const for some s an 0, then ux, t u x, t L 1 x C T ux, 0 u x, 0 L 1 x + C s,t s/4+1/θ log + C s,t s+1. Proof. The theorem follows irectly from Theorem.1in [15], semiconcave stability Lemma 4., an the truncation error estimate Lemma 5.1. Remark. The work is unerway on the spectral viscosity metho for the initialbounary Hamilton Jacobi problem with nonperioic bounary conitions. Acknowlegments. I woul like to express my eep gratitue to Chi-Wang Shu for helpful suggestions uring the preparation of this paper. I am also greatly inebte to Eitan Tamor an to unknown referees for comments on a raft of this paper. I wish to thank Jan Hesthaven an Jing Shi for stimulating conversations, which motivate the present stuy. REFERECES [1] R. Abgrall, umerical iscretization of the first-orer Hamilton Jacobi equation on triangular meshes, Comm. Pure Appl. Math., , pp [] R. A. Aams, Sobolev Spaces, Acaemic Press, San Diego, [3] T. Barth an J. Sethian, umerical schemes for the Hamilton Jacobi an level set equations on triangulate omains, J. Comput. Phys., , pp [4] C. Canuto, M. Y. Hussaini, A. Quarteroni, an T. Zang, Spectral Methos in Flui Dynamics, Springer-Verlag, ew York, [5] G.-Q. Chen, Q. Du, an E. Tamor, Spectral viscosityapproximations to multiimensional scalar conservation laws, Math. Comp., , pp [6] M. G. Cranall an P. L. Lions, Viscositysolutions of Hamilton Jacobi equations, Trans. Amer. Math. Soc., , pp. 1 4.

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