ON NUMERICAL METHODS AND ERROR ESTIMATES FOR DEGENERATE FRACTIONAL CONVECTION-DIFFUSION EQUATIONS

Transcription

1 ON NUMERICAL MEHODS AND ERROR ESIMAES FOR DEGENERAE FRACIONAL CONVECION-DIFFUSION EQUAIONS SIMONE CIFANI AND ESPEN R. JAKOBSEN Abstract. First we introduce and anayze a convergent numerica method for a arge cass of noninear nonoca possiby degenerate convection diffusion equations. Secondy we deveop a new Kuznetsov type theory and obtain genera and possiby optima error estimates for our numerica methods even when the principa derivatives have any fractiona order between 1 and! he cass of equations we consider incudes equations with noninear and possiby degenerate fractiona or genera Levy diffusion. Specia cases are conservation aws, fractiona conservation aws, certain fractiona porous medium equations, and new strongy degenerate equations. 1. Introduction In this paper we deveop a numerica method aong with a genera Kuznetsov type theory of error estimates for integro partia differentia equations of the form { t u + divfu = L 1.1 µ [Au], x, t, ux, 0 = u 0 x, x R d, where = R d 0, and the nonoca diffusion operator L µ is defined as 1. L µ [φ]x = φx + z φx z φx 1 z <1 z dµz, z >0 for smooth bounded functions φ. Here 1 denotes the indicator function. hroughout the paper the data f, A, µ, u 0 is assumed to satisfy: A.1 f = f 1,..., f d W 1, R; R d with f0 = 0, A. A W 1, R, A non-decreasing with A0 = 0, A.3 µ 0 is a Radon measure such that z >0 z 1 dµz <, A.4 u 0 L R d L 1 R d BV R d. We use the notation a b = mina, b and a b = maxa, b. Remark 1.1. hese assumptions can be reaxed in two standard ways: i f, A can take any vaue at u = 0 repace f by f f0 etc., and ii f, A can be assumed to be ocay Lipschitz. By the maximum principe and A.4, soutions of 1.1 are bounded, and ocay Lipschitz functions are Lipschitz on compact domains. he measure µ and the operator L µ are respectivey the Lévy measure and the generator of a pure jump Lévy process. Any such process has a Lévy measure Key words and phrases. Fractiona conservation aws, convection-diffusion equations, porous medium equation, entropy soutions, numerica method, convergence rate, error estimates. his research was supported by the Research Counci of Norway NFR through the project Integro-PDEs: numerica methods, anaysis, and appications to finance. 1

2 S. CIFANI AND E. R. JAKOBSEN and generator satisfying 1. and A.3, see e.g. [4]. Exampe are the symmetric α-stabe processes with fractiona Lapace generators where dz 1.3 dµz = c λ z d+λ c λ > 0 and L µ λ/ for λ 0,. Non-symmetric exampes are popuar in mathematica finance, e.g. the CGMY mode where C e G z dz for z > 0, z 1+λ dµz = C e M z dz for z < 0, z 1+λ and where d = 1, λ= Y 0,, and C, G, M > 0. We refer the reader to [14] for more detais on this and other nonoca modes in finance. In both exampes the nonoca operator behaves ike a fractiona derivative of order between 0 and. Equation 1.1 has a oca non-inear convection term the f-term and a fractiona or nonoca non-inear possiby degenerate diffusion term the A-term. Specia cases are scaar conservation aws A 0, fractiona and Lévy conservation aws Au = u and α-stabe or more genera µ see e.g. [6, 1] and [7, 9, 5], fractiona porous medium equations [16] A = u m 1 u for m 1 and α-stabe µ, and strongy degenerate equations where A vanishes on a set of positive measure. If either A is degenerate or L µ is a fractiona derivative of order ess than 1, then soutions of 1.1 are not smooth in genera and uniqueness fais for weak distributiona soutions. Uniqueness can be regained by imposing additiona entropy conditions in a simiar way to what is done for conservation aws. he Kruzkov entropy soution theory of scaar conservation aws [7] was extended to cover fractiona conservation aws in [1], to more genera Lévy conservation aws in [5], and then finay to setting of this paper, equations with non-inear fractiona diffusion and genera Lévy measures in [11]. For oca nd order degenerate convection diffusion equations ike 1.4 t u + divfu = Au, there is an entropy soution theory due to Carrio [9]. In recent years, integro partia differentia equations ike 1.1 have been at the center of a very active fied of research. A thorough description of the mathematica background for such equations, reevant bibiography, and appications to severa discipines of interest can be found in [1,, 7, 11, 16, 5]. he first contribution of this paper is to introduce a numerica method for equation 1.1 and prove that it converges toward the entropy soution of 1.1 under assumptions A.1 A.4. he numerica method is based upon a monotone finite voume discretization of an approximate equation with truncated and hence bounded Lévy measure. Essentiay it is an extension of the method in [11] from symmetric α-stabe to genera Lévy measures, but since non-symmetric measures are aowed, the discretization becomes more compicated here. Apart from its abiity to capture the correct soution for the whoe famiy of equations of the form 1.1, the main advantage of our numerica method is that it aows for a compete error anaysis through the new framework for error estimates that we deveop in the second part of the paper. he second, and probaby most important contribution of the paper, is the deveopment of a theory capabe of producing error estimates for degenerate equations of order greater than 1. his theory is based on a non-trivia extension of the Kuznetsov theory for scaar conservation aws [8] to the current fractiona diffusion setting. An initia step in this anaysis was performed in [], with the derivation of a so-caed Kuznetsov emma in a reevant form for 1.1. In [] the emma is

3 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 3 used in the derivation of continuous dependence estimates and error estimates for vanishing viscosity type of approximations of 1.1. In the present paper, we show how it can be used in soving the more difficut probem of finding error estimates for numerica methods for 1.1. As a coroary of our Kuznetsov type theory, we obtain expicit λ-dependent error estimates when µ is a measure satisfying z <1 dµz c λ dz z d+λ for c λ > 0 and λ 0,. In this paper we wi ca such measures fractiona measures. For exampe for the impicit version of our numerica method 3.5, we prove in Section 6 that 1 λ 0, 1, u, u, L1 R d C 1 og λ = 1, λ λ 1,, where u is the entropy soution of 1.1 and u is the soution of 3.5. Note that our error estimate covers a vaues λ 0,, a spacia dimensions d, and possiby strongy degenerate equations! Aso note that under our assumptions, the soution u possiby ony have BV reguarity in space. Hence the error estimate is robust in the sense that it hods aso for discontinuous soutions, and moreover, the cassica resut of Kuznetsov [8] for conservation aws foows as a coroary by taking A 0 a vaid choice here! and λ 0, 1. he above estimate is aso consistent with error estimates for the vanishing λ-fractiona viscosity method, t u + divfu = λ/ u as 0 +, see e.g. [18, 1], but note that our probem is different and much more difficut. here is a vast iterature on approximation schemes and error estimates for scaar conservations aws, we refer e.g. to the books [6, ] and references therein for more detais. For oca degenerate convection-diffusion equations ike 1.4, some approximation methods and error estimates can be found e.g. in [0, 1, 4] and references therein. In this setting it is very difficut to obtain error estimates for numerica methods, and the ony resut we are aware of is a very recent one by Karsen et a. [4] but see aso [10]. his very nice resut appies to rather genera equations of the form 1.4 but in one space dimension and under additiona reguarity assumptions e.g. x Au BV. When it comes to nonoca convection-diffusion equations, the iterature is very recent and not yet very extensive. he paper [15] introduce finite voume schemes for radiation hydrodynamics equations, a mode where L µ is a nonoca derivative of order 0. hen fractiona conservation aws are discretized in [17, 13, 1] with finite difference, discontinuous Gaerkin, and spectra vanishing viscosity methods respectivey. In [15, 13] Kuznetsov type error estimates are given, but ony for integrabe Lévy measures or measures ike 1.3 with λ < 1. Both of these resuts can be obtained through the framework of this paper. In [1] error estimates are given for a λ but with competey different methods. he genera degenerate non-inear case is discretized in [11] without error estimates for symmetric α-stabe Lévy measures and then in the most genera case in the present paper. Linear non-degenerate versions of 1.1 frequenty arise in Finance, and the probem of soving these equations numericay has generated a ot of activity over the ast decade. An introduction and overview of this activity can be found in the book [14], incuding numerica schemes based on truncation of the Lévy measure. We aso mention the iterature on fractiona and nonoca fuy non-inear equations ike e.g. the Beman equation of optima contro theory. Such equations have been intensivey studied over the ast decade using viscosity soution methods, incuding

4 4 S. CIFANI AND E. R. JAKOBSEN initia resuts on numerica methods and error anaysis. We refer e.g. [5, 8, 3] and references therein for an overview and the most genera resuts in that direction. In fact, ideas from that fied has been essentia in the deveopment of the entropy soution theory of equations ike 1.1, and the construction of monotone numerica methods of this paper paraes the one in [8]. However the structure of the two casses of equations aong with their mathematica and numerica anaysis are very different. his paper is organized as foows. In Section we reca the entropy formuation and we-posedness resuts for 1.1 of [11] and the Kuznetsov type emma derived in []. We present the numerica method in Section 3. here we focus on the case of no convection f 0 to simpify the exposition and focus on new ideas. In Section 4 we prove severa auxiiary properties of the numerica method which wi be usefu in the foowing sections. We estabish existence, uniqueness, and a priori estimates for the soutions of the numerica method in Section 5. he genera Kuznetsov type theory for deriving error estimates is presented in Section 6, where it is aso used to estabish a rate of convergence for equations with fractiona Lévy measures, i.e. 1.5 hods. In Section 7 we extend a the resuts considered so far to genera convection-diffusion equations of the form 1.1 with f 0. Finay, we give the proof of the main error estimate heorem 6.1 in Section 8.. Preiminaries In this section we briefy reca the entropy formuation for equations of the form 1.1 introduced in [11], and the new Kuznetsov type of emma estabished in []. Let ηu, k = u k, η u, k = sgn u k, q u, k = η u, k f u f k for = 1,..., d, and write the nonoca operator L µ [φ] as where L µ r [φ]x = L µ,r [φ]x = γ µ,r L µ r [φ] + L µ,r [φ] + γ µ,r φ, 0< z r z >r = z >r φx + z φx z φx1 z 1 dµz, φx + z φx dµz, z 1 z 1 dµz, = 1,..., d. We aso define µ by µ B = µ B for a Bore sets B 0. Let us reca that ϕx L µ [ψ]x dx = ψx L µ [ϕ]x dx R d R d for a smooth L L 1 functions ϕ, ψ, cf. [, 11]. Definition.1. Entropy soutions A function u L C[0, ]; L 1 R d is an entropy soution of 1.1 if, for a k R, r > 0, and test functions 0 ϕ C c R d [0, ],.1 ηu, k t ϕ + qu, k + γ µ,r ϕ + ηau, Ak L µ r [ϕ] + η u, k L µ,r [Au] ϕ dx dt ηux,, k ϕx, dx + ηu 0 x, k ϕx, 0 dx 0. R d R d Note that γ µ,r 0 when the Lévy measure µ is symmetric, i.e. when µ µ. From [11] we now have the foowing we-posedness resut.

5 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 5 heorem.1. We-posedness Assume A.1 A.4 hod. hen there exists a unique entropy soution u of 1.1 such that u L C[0, ]; L 1 R d L 0, ; BV R d, and the foowing a priori estimates hod u, t L R d u 0 L R, d u, t L1 R d u 0 L1 R, d u, t BV R d u 0 BV R, d u, t u, s L 1 R d σ t s, for a t, s [0, ] where { c r if z 1 dµz <, z >0 σr = c r 1 otherwise. Moreover, if aso 1.5 hods, then c r if λ 0, 1, σr = c r n r if λ = 1, c r 1 λ if λ 1,. he ast a priori estimate is sighty more genera then the one in [11], and foows e.g. in the imit from the estimates in Lemmas 5.3 and 5.4. We now reca the new Kuznetsov type of emma estabished in []. Let ω C c R, 0 ω 1, ωτ = 0 for a τ > 1, and R ωτ dτ = 1, and define ω δ τ = 1 δ ω τ δ, Ω x = ω x 1 ω x d, and ϕ,δ x, y, t, s = Ω x y ω δ t s for, δ > 0. We aso need. E δ v = sup v, t v, s L1 R. d t s <δ t,s [0, ] In the foowing we et dw = dx dt dy ds and C 0 be a constant depending on time and the initia data u 0 that may change from ine to ine. Lemma.. Kuznetsov type of emma Assume A.1 A.4 hod. Let u be the entropy soution of 1.1 and v be any function in L C[0, ]; L 1 R d

6 6 S. CIFANI AND E. R. JAKOBSEN L 0, ; BV R d with v, 0 = v 0. hen, for any, r > 0 and 0 < δ <, u, v, L1 R d u 0 v 0 L1 R d + C + E δ u E δ v ηvx, t, uy, s t ϕ,δ x, y, t, s dw Q qvx, t, uy, s x ϕ,δ x, y, t, s dw Q + ηavx, t, Auy, s L µ r [ϕ,δ x,, t, s]y dw Q η vx, t, uy, s L µ,r [Av, t]x ϕ,δ x, y, t, s dw Q ηavx, t, Auy, s γ µ,r x ϕ,δ x, y, t, s dw Q + ηvx,, uy, s ϕ,δ x,, y, s dx dy ds R d ηv 0 x, uy, s ϕ,δ x, 0, y, s dx dy ds R d he proof is given in []. he origina resut of resut of Kuznetsov in [8] is a specia case when µ = 0 or A = he numerica method In this section we derive our numerica method. Here and in the foowing sections we focus on the case f 0 to simpify the exposition and focus on the new ideas. he genera case f 0 wi then be treated at the end, in Section 7. We wi consider uniform space/time grids given by x α = α for α Z d and t n = n t for n = 0,..., N = t. We aso use the foowing rectanguar subdivisions of space = x α + 0, 1 d for α Z d. We start by discretizing the nonoca operator, repacing the measure µ by the bounded truncated measure 1 z > zµ and the gradient by a numerica gradient 3.1 D = D 1,, D d, where D D γ are upwind finite difference operators defined by D + 3. D γ φx = φx := φx + e φx µ, for γ > 0, D φx := φx φx e otherwise. Here e 1,..., e d is the standard basis of R d. his gives an approximate nonoca operator L µ [Aφ]x 3.3 µ, = Aφx + z Aφx dµz + γ D Aφx, z > which is monotone by upwinding and non-singuar since the truncated measure is bounded. A semidiscrete approximation of 1.1 with f 0 is then obtained by soving the approximate equation 3.4 t u = L µ [Au],

7 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 7 by a finite voume method on the spacia subdivision { } α. I.e. for each t, we ook for piecewise constant approximate soution Ux, t = Z d U t 1 R x, that satisfy 3.4 in weak form with 1 1 d R as test functions: For every α Z d, 1 d t U dx = 1 d Lµ [AU] dx. Finay we discretize in time by repacing t by backward or forward differences D ± t and U αt by a piecewise constant approximation U n α. he resut is the impicit method 3.5 U n+1 α = U n α + t L µ AU n+1 α, and the expicit method 3.6 where U n+1 α = U n α + t L µ AU n α L µ AU n α = 1 d Lµ [AŪ n ]x dx, and Ū n x = Z U n d 1 R x is a piecewise constant x-interpoation of U. As initia condition for both methods we take Uα 0 = 1 d u 0 x dx for a α Z d. Lemma 3.1. L µ AU n α = Z G α AU n with G α = G α, + G α, and 3.7 G α, = 1 d 1 R x + z 1 R x dµz dx, z > d G α, µ, 1 = γ d D γ 1 R x dx. =1 Remark 3.. G α, is a oepitz matrix cf. Lemma 4.1 b whie G α, is a tridiagona matrix. When the measure µ is symmetric, then G α, is symmetric and G α, = 0. Proof. Since AŪx = Z d AU 1 R x and D Ūx = Z d U D 1 R x,

8 8 S. CIFANI AND E. R. JAKOBSEN we find that d Lµ AU α = = z > Lµ [AŪ]x dx AŪx + z AŪx dµz dx µ, + γ AU D 1 R x dx Z d = AU 1 R x + z 1 R x dµz dx Z d z > + d µ, AU γ D γ 1 R x dx. Z d =1 he proof is compete. 4. Properties of the numerica method In this section we show that the numerica methods are conservative, monotone and consistent in the sense that certain ce entropy inequaities are satisfied. We start by a technica emma summarizing the properties of the weights G α defined in 3.7. Lemma 4.1. a α Z d Gα = α Z d G α = 0 for a Z d. b G α = G +e α+e for a α, Z d and = 1,..., d. c G 0 and Gα 0 for α. d here is c = cd, µ > 0 such that G σ µ and where { s when z 1 dµz <, 4.1 σ µ s = s otherwise. e If 1.5 hods, then there is c = cd, λ > 0 such that G s λ for λ > 1, s 4. σ λ s = n s for λ = 1, s for λ < 1. c c σ λ for Proof. a By the definitions of G α,, G α, and Fubini s theorem, d G α, = 1 R x + z dx 1 R x dx dµz = 0, α Z d R d R d d α Z d G α, = ± z > d γ =1 µ, 1 R x ± e dx 1 R xdx = 0, R d R d and, since Z 1 d R x 1, d G α, = 1 R x + z 1 R x dµz dx = 0, Z d z > Z d Z d d µ, d G α, γ = ± 1 R x ± e 1 R x dx = 0. Z d =1 Z d Z d

9 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 9 herefore α Z d Gα = α Z d Gα, + G α, = 0 and Z d Gα = 0. b Let y = x + e and note that d G α, = 1 R y e + z 1 R y e dµz dy = +e +e z > z > = d G +e,α+e. In a simiar fashion we get G α, = G +e,α+e. c Note that d G, = R 1 R+e y + z 1 R+e y dµz dy z > 1 R x + z 1 dµz dx 0. whie by the definition D γ, see 3., For α, d G, = d γ =1 d G α, = µ, sgn γ z > µ, G α, = 0 for α ± e, and by the definition of D γ, R 1 R x 1 R x + z dµz dx 0, dx 0. d G ±e, = d γ =1 µ, sgn γ µ, R±e 1 R x ± e dx 0. herefore G = G, + G, 0 and G α = G α, + G α, 0 for α. d o find the ower bound on G we note that R 1 R x+z 1 R x dx d, and hence z G, dµz 1 z <1z + 1 z >1z dµz. z <1 z > he bound then foows since G, d z dµz d < z <1 0< z <1 z dµz. When z 1 dµz <, the corresponding bound foows by a simiar argument. e When 1.5 hod we can estimate G, in the foowing way z c λ dz G, < z <1 z d+λ dµz z >1 σ c d λ 1 λ 1 1 λ + C for λ 1, = c λ σ d n + C for λ = 1.

10 10 S. CIFANI AND E. R. JAKOBSEN he ast equaity can be proved using poar coordinates, and σ d is the surface area of the unit sphere in R d. Simiary we find that G, c λ dz σ d d z < z <1 z d+λ = dc 1 λ 1 1 λ for λ 1, λ σ d n for λ = 1, and since 1 1 λ is ess than 1 or 1 λ when λ < 1 or λ > 1 respectivey and when <, the proof is compete. From the two facts that G α 0 when α and sgnuau = Au, we now immediatey get a Kato type inequaity for the discrete nonoca operator 3. Lemma 4.. Discrete Kato inequaity If {u α, v α } α Z d are two bounded sequences, then sgnu α v α G α Au Av G α Au Av. Z d Z d From Lemma 4.1 it aso foows that the expicit method 3.6 and the impicit method 3.5 are conservative and monotone, at east when the expicit method satisfies the foowing CFL condition: t 4.3 cl A σ µ < 1 where σ µ is defined in 4.1. Here c is defined in Lemma 4.1, and L A denotes the Lipschitz constant of A. When the Lévy measure µ aso satisfies 1.5, we have a weaker CFL condition 4.4 t cl A σ λ < 1 where σ λ is defined in 4.. Proposition 4.3 Conservative monotone schemes. a he impicit and expicit methods 3.5 and 3.6 are conservative, i.e. for an 1 -soution U, Uα n = Uα. 0 α α b he impicit method is monotone, i.e. if U and V sove 3.5, then U n V n U n+1 V n+1 for n 0. c If 4.3 or 4.4 and 1.5 hods, then the expicit method 3.6 is monotone. t Remark 4.4. he CFL condition 4.3 impies that C in genera just as for the heat equation, and t C when z 1 dµz <. Condition 4.3 is sufficient for a equations considered in this paper. In rea appications however, typicay 1.5 hods, and the superior CFL condition 4.4 shoud be used. Proof. a Sum 3.5 or 3.6 over α, change the order of summation, and use Lemma 4.1 a: Uα n+1 = Uα n + t AU G α = Uα n. α Z d α Z d Z d α Z d α Z d c Let α [u] = u α + t Z d Gα Au, the right hand side of 3.6. By Lemma 4.1 c, G α 0 for α and hence u α [u] 0 for α. Since A non-decreasing and G α α 0, we use the ower bound on G α α in Lemma 4.1 c to find that uα α [u] = 1 + t G α α A t u α 1 cl A σ µ,

11 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 11 which is positive by the CFL condition 4.3. b he proof is simiar to and easier than the proof of c. We then turn to checking the consistency of the method, and to do that we write G α, = G r α, + G α,,r and G α, = G α,,r + G α, r for r > 0 where G r α, = 1 d 1 R x + z 1 R x dµz dx, < z r G α,,r = 1 d 1 R x + z 1 R x dµz dx, G α,,r = 1 d for γ G α, r = 1 d d γ =1 µ,,r d =1 z >r µ,,r = γ µ,r D γr 1 R x dx < z r z 1 z 1 dµz, D γ r 1 R x dx. If r <, we set Gr α, = 0 = Gα,,r. We aso define G,r α = G r α, + G α,,r and G α,r = G α,,r + G α, r, and note that Lemmas 4.1 and 4. obviousy sti hods with G,r α G α. or G α,r repacing Proposition 4.5. Ce-entropy inequaities a If U is a soution of the impicit method 3.5, then, for a r > 0 and k R, ηuα n+1, k ηuα n, k + t G α,r ηau n+1, Ak Z d t η Uα n+1, k G α,r AU n+1. Z d b Assume the CFL condition 4.3 or 4.4 and 1.5 hods. If U is a soution of the expicit method 3.6, then, for a r > 0 and k R, ηuα n+1, k ηuα n, k + t G α,r ηauα n, Ak Z d t η Uα n+1, k G α,r AUα n. Z d Remark 4.6. In the ce-entropy inequaity for the expicit method, the η -term appears in the wrong time. In Section 6, we wi see that this eads to worse error estimates for the expicit method than for the impicit method. Remark 4.7 Convergence to entropy soutions. Proposition 4.5 and a standard argument show that any C[0, ]; L 1 oc Rd -convergent sequence of interpoated soutions ū of 3.5 or 3.6, wi converge to an entropy soution of 1.1. We refer to heorem 3.9 in [] and Section 4. in [11] for more detais. Convergence to the entropy soution aso foows from the error estimates of Section 6. Proof. a By 3.5 we easiy see that for any k R, U n+1 α k U n α k + t 1 k,+ U n+1 α L µ AU n+1 α, U n+1 α k U n α k + t 1,k U n+1 α L µ AU n+1 α.

12 1 S. CIFANI AND E. R. JAKOBSEN Subtracting and using ηu, k = u k and η u, k = sgn u k, we find that ηu n+1 α, k ηu n α, k + t η U n+1 α, k L µ AU n+1 α. For any r > 0, we use Lemmas 4.1 a and 4. with G α,r η Uα n+1, k G α,r AU n+1 Z d = η Uα n+1, k G α,r Z d ηau n+1, Ak. Z d G α,r AU n+1 Ak repacing G α to see that since Z d G α,r he ce entropy inequaity now foows from writing G α = Gα,r + Gα,r the above inequaities. = 0 and using b By 3.6 and monotonicity Proposition 4.5 c we obtain the foowing inequaities: For a r > 0, k Uα n k + t U n+1 α U n+1 α Z d G α,r AU n k + t 1 k,+ Uα n+1 G α,r AU n, Z d k Uα n k + t G α,r AU n k Z d + t 1,k Uα n+1 G α,r AU n. Z d Since ηau, Ak = AU k AU k, the ce entropy inequaity foows from subtracting the two inequaities. 5. A priori estimates, existence, and uniqueness In this section we state and prove severa a priori estimates for the soutions of the numerica methods 3.5 and 3.6. In what foows, we wi use different interpoants ū of the soutions U n α of the schemes. For the impicit method 3.5 we take 5.1 whie for the expicit method 3.6, ūx, t = U n+1 α for a x, t t n, t n+1 ], ūx, t = U n α for a x, t [t n, t n+1. We now prove the foowing a priori estimates for ū: ū, t L1 R d u 0 L1 R d, ū, t L R d u 0 L R d, ū, t BV R d u 0 BV R d. Lemma 5.1. A priori estimates a If U sove 3.5 and ū is defined by 5.1, then the a priori estimates hod for a t > 0. b Assume the CFL condition 4.3 or 4.4 and 1.5 hods. If U sove 3.6 and ū is defined by 5., then the a priori estimates hod for a t > 0.

13 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 13 Proof. Since the schemes are conservative and monotone, cf. Proposition 4.3, this is a standard resut that essentiay foows from the Cranda-artar Lemma. For expicit methods in part b we refer to e.g. heorem 3.6 in [] for the detais. We did not find a reference for impicit methods, so we give a proof of part a here. See aso [17] for the case when A is inear. Let u α = Uα n+1, h α = Uα n, and write 3.5 as 5.6 u α t G α Au = h α. Z d We prove 5.3. Mutipy 5.6 by sgnu α and use Lemma 4. to get u α t Z d G α Au h α, which by Fubini s theorem and the fact that α Z d Gα = 0 impies that u α h α. α Z d α Z d By the definition of u α, h α and an iteration in n, it foows that Uα n Uα. 0 α Z d α Z d By 5.1, ū, t L1 R d = d α Z d U n α for t t n, t n+1 ], and 5.3 foows. o prove 5.5, we subtract two equations 5.6 evauated at different points, u α u α e t G α Au G α e Au = h α h α e Z d and use the fact that G α = Gα+e +e to see that u α u α e t Au Au e = h α h α e. Z d G α hen we mutipy by sgnu α u α e, use Lemma 4., and sum over α, to find that u α u α e h α h α e. α Z d α Z d he estimate 5.5 then foows by iteration and the definitions of u α, h α, ū. It remains to prove 5.4. Note that since α u α < by 5.3, there is an α 0 such that sup α u α = u α0. Moreover, the paraboic term is nonpositive at the maximum point: since Z d Gα = 0 and Z d Gα <, G α0 Z d Au = G α0 Z d Au Au α0 0. hen by the above inequaity and 5.6, sup u α = u α0 u α0 t G α0 Au = h α0 sup h α. α Z d Z d α Z d In a simiar way we find that inf α Z d h α inf α Z d u α and 5.4 foow from the definitions of u α, h α, ū and an iteration in n. Lemma 5. Goba existence and uniqueness. a here exists a unique soution U n 1 of the impicit scheme 3.5 for a n 0. b Assume the CFL condition 4.3 or 4.4 and 1.5 hods. hen there exists a unique soution U n 1 of the expicit scheme 3.6 for a n 0.

14 14 S. CIFANI AND E. R. JAKOBSEN Note that U n 1 impies that ū, t L 1 R d. Proof. a Let u α = Uα n+1 and h α = Uα n, rewrite 3.5 as 5.6, define α [u] = u α u α t G α Au h α, Z d and et be such that G α t 1 + L A c < 1. σ µ We first show that α is monotone, i.e. u v impies α [u] α [v]. For α, 0 by Lemma 4.1, and hence since A non-decreasing, u α [u] 0. Moreover, since A non-decreasing and σ µ Gα α 0, uα α [u] = 1 + t G α α A t u α L A c σ µ which is positive by our choice of. Since is monotone and A is nondecreasing, + α [u] α [v] α [u v] α [v] α = 1 α Z d u α v α v α + t α Z d = 1 u α v α + + t α Z d α Z d c Z d G α α Z d G α Au v Av Au Av +. A simiar estimate hods for α α[u] α [v], and since α Z d Gα = 0, we have shown that α [u] α [v] 1 u α v α. α Z d α Z d So α is an 1 -contraction and Banach s fixed point theorem then impies that there exists a unique soution ū 1 of α [ū] = ū α and hence aso of 5.6. b Existence foows by construction and the a priori estimates in Lemma 5.1. Uniqueness essentiay foows by monotonicity and α Gα = 0: Assume two soutions U n and V n, subtract the two equations and mutipy by sgn U n V n, and use the Kato inequaity Lemma 4. aong with α Gα = 0 to show that α U n V n α U 0 V 0. We have the foowing reguarity estimate in time: Lemma 5.3. Reguarity in time a Assume A. A.4 hod, and et U be a soution of the impicit method 3.5 and ū defined by 5.1. hen ū, s ū, t L1 R d σ µ s t + t for a s, t > 0, where r if z 1 dµz <, z >0 σ µ r = r otherwise.

15 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 15 b Assume A. A.4 and 4.3 or 4.4 and 1.5 hod, and et U be a soution of the expicit method 3.6 and ū defined by 5.. hen ū, s ū, t L 1 R d σ µ s t + t for a s, t > 0, where σ µ is defined in a. Proof. he two proofs are essentiay identica, so we ony do the proof for case a. 1 By 3.5, we find that for any x, Uα n Uα n 1 = t d L[A Ū n ]x dx. ake a test function 0 φ Cc and define φ α = 1 R d α φydy and φx = α φ α1 Rα x. Mutipy the equation by d φ α and sum over α to find that φx Ū n x Ū n 1 x dx = t φx L[A Ū n ]x dx, R d where Let L be the adjoint of L, then since Ū is constant over, φxū n x Ū n 1 x dx = φx Ū n x Ū n 1 x dx R d R d = t φx φx L[AŪ n ]x dx + t L [φ]xaū n x dx. R d R d Let ω ε be an approximate unit, i.e. ω ε x = 1 ω x ε d ε where 0 ω C 0 and ω dx = 1. ake φx = ω R d ε y x in the equation above and et Uε n = Ū n ω ε : Ū n ε Ū n 1 ε = t ω ε ω ε L[AŪ n ] + t L [ω ε ] AŪ n. By Fubini we then find that 1 t Ū n ε Ū n 1 ε L 1 ω ε ω ε L 1 L[AŪ n ] L 1 + L [ω ε ] AŪ n L 1 = I 1 + I. R d 3 o estimate I 1, note that by a standard argument ω ε ω ε L 1 ω ε BV = c ω ε, and then by the definition of L in 3.3, Fubini, the L 1 BV reguarity of U n Lemma 5.1, and the reguarity of A in A., L[AŪ n ] L 1 = AŪ n x + z AŪ n x z D AŪ n x1 z <1 dx dµz z > R d AU n BV z 1 z <1 + AU n L 11 z >1 dµz C z > z > z 1 dµz C z >0 z 1 dµz. hese estimates aong with A.3 shows that I 1 Cε 1.

16 16 S. CIFANI AND E. R. JAKOBSEN 4 hen we estimate I. Note first that since D = D + D D, we can use ayor s formua to see that φx + z φx z D φx = sz D φx + szz ds ± d 1 z i 1 sφ xix i x ± s ds. his identity aong with the definition of L, repeated use of Fubini, and one integration by parts in x, then eads to L [ω ε ] AŪ n x 1 = 1 sdω ε x y + szz z DAŪ n y dy dµz ds 0 < z <1 R d d 1 1 s xi ω ε x y ± sz i xi AŪ n y dydµzds i=1 0 < z <1 R d + ω ε x y + z ω ε x y AŪ n x dµz dy. R d z >1 Here DAŪ n y dy shoud be interpreted as a measure, and DAŪ n y dy = d AU n y = AU n BV. By Young s inequaity for convoutions Fubini in our case, we then find that I 3 ω ε BV AŪ n BV Cε 1. 0< z <1 i=1 0 z dµz + ω ε L 1 AŪ n L 1 z >1 Here again we have used the properties and reguarity of µ, A, Ū n, and ω ε. 5 By steps 4 we can concude that Ū ε n Ū ε m L 1 n j=m+1 Ū ε j Ū ε j 1 L 1 C n m t, ε dµz where the constant C does not depend on n or m. By the triange inequaity and standard BV -estimates, it then foows that Ū n Ū m L 1 Ū n Ū n ε L 1 + Ū n ε Ū m ε L 1 + Ū m ε Ū m L 1 and hence by taking ε = C n m t, Ū n BV ε + C ε n m t + Ū m BV ε, Ū n Ū m L 1 C n m t. For the time-interpoated function ū defined in 5.1, we then find the foowing estimate ū, t ū, s L 1 = Ū n Ū m L 1 C n m t C t s + t. he equaity foows since for each t, s there are n, m such that ūx, t = Ū n x and ūx, s = Ū m x. Moreover, by the definition of ū, n m t t s + t. It remains to prove a better estimate for the case when z 1 dµz <. his proof is simiar but much easier than the proof above, so we skip it.

17 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 17 he time reguarity resut in Lemma 5.3 is not optima for Levy operators L with order in the interva [1,. o get optima resuts we need more detaied information on the Levy measure µ than merey assumption A.3. We wi now prove an improved time reguarity resut for fractiona measures 1.5. In this resut we wi need the foowing CFL condition, 5.7 C t < 1 for λ 0,. 1 λ Lemma 5.4. ime reguarity for fractiona measures Assume A. A.4, and 1.5 hod. a If the CFL condition 5.7 hod, U is a soution of the impicit method 3.5 and ū its interpoation defined by 5.1, then for a s, t > 0, ū, s ū, t L1 R d σ λ s t + t; σ λ τ = τ λ < 1, τ n τ λ = 1, τ 1 λ λ > 1. b If the CFL condition 4.4 hod, U is a soution of the expicit method 3.6 and ū its interpoation defined by 5., then for a s, t > 0, τ λ < 1, ū, s ū, t L 1 R d σ λ s t + t; σ λ τ = τ α for any α 0, 1 λ = 1, τ 1 λ λ > 1. Note we that in this resut we need the CLF condition aso for the impicit scheme. he reason is that the time-reguarity is inked through the equation to the approximate -depending diffusion term as wi be seen from the proof. For the impicit scheme, we can have better resuts for λ = 1 since we can use the ess restrictive CFL condition 5.7. Proof. he resut for λ < 1 is a coroary to Lemma 5.3. he proof for λ 1 is the same as the proof of Lemma 5.3, except that we use different estimates for I 1 and I in step. From step 3 in that proof and 1.5 and a simpe computation in poar coordinates, we get that I 1 C ε { C ε z > z 1 dµz C ε + λ for λ > 1, n for λ = 1. z dz < z <1 z d+λ + C o estimate I, we use ayor expansions and integration by parts to find that L [ω ε ] AŪ n x = 1 1 sdω ε x y + szz z DAŪ n y dy dµz ds 0 < z <ε R d 1 ω ε x y + sz ω ε x y z DAŪ n y dy dµz ds 0 d i=1 + R d ε< z <1 1 d i= z >1 R d ε< z <1 < z <ε R d R d 1 s xi ω ε x y ± sz i xi AŪ n y dydµzds ω ε x y ± s ω ε x y z i xi AŪ n y dydµzds ω ε x y + z ω ε x y AŪ n x dµz dy.

18 18 S. CIFANI AND E. R. JAKOBSEN hen by Fubini, the definition of ω ε, and the change of variabes x, z εx, εz, Dω ε x + sz z dµz c λ ε 1 λ z dz Dω dx R d < z <ε R d 0< z <1 z d+λ. By simiar estimates and Young s inequaity for convoutions we find that I c λ ε 1 λ AŪ n BV 3 ω BV C + AŪ n L 1 ω ε L 1 { ε 1 λ + 1, λ > 1, n ε + 1, λ = 1. z >1 0< z <1 dµz z dz z d+λ + 4 ω L 1 Note that the n ε-term comes from the integra over 1 < z < 1 ε. As in step 5 in the proof of Lemma 5.3, we then find that 1< z < 1 ε Ū n Ū m L 1 Ū n BV ε + n m ti 1 + I + Ū m BV ε. o concude, we assume that ε which means in particuar that { ε 1 λ + 1, λ > 1, I 1 + I C n ε + 1, λ = 1. z dz z d+λ When λ > 1, the fina resut foows from taking ε = c n m t 1 λ and arguing as in the end of the proof of Lemma 5.3. Note that in view of the CFL conditions 4.4 and 5.7, the constant c can be chosen such that ε. For λ = 1, we can use ε = c n m t for the impicit method in view of 5.7, and by 4.4, ε = c n m t α for any α 0, 1, wi do the job for the expicit method. By the a priori estimates Lemma 5.1 and 5.3 and Komogorov s compactness theorem cf. e.g. [, heorem 3.8], we find subsequences of both methods 3.5 and 3.6 converging to some function u. he function u inherits a the a priori estimates of ū, and it wi be the unique entropy soution of 1.1 by Remark 4.7. In short, we have the foowing resut: heorem 5.5. Compactness Assume A. A.4 hod. If either i U is the soution of the impicit method 3.5 and ū defined by 5.1, or ii U is the soution of the expicit method 3.6, ū defined by 5., and 4.3 or 4.4 and 1.5 aso hods, then there is a subsequence of {ū} >0 converging in C[0, ]; L 1 R d to the unique entropy soution u of 1.1 as 0. Moreover, u L C[0, ]; L 1 R d L 0, ; BV R d. Remark 5.6. his resut provides a proof for the existence resut heorem 5.3 in [11] for L 1 L BV entropy soutions of 1.1, and then the genera existence resut in L 1 L foows by a density argument using the L 1 -contraction. 6. Error estimates In this section we give different error estimates and convergence resuts for our schemes, estimates that are vaid for genera Levy measures and better estimates

19 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 19 that hods for fractiona measures satisfying 1.5. o give the genera resut, we need the foowing quantities: I,r 1 = 1 z dµz, z r I,δ,r = + t z dµz + dµz ; δ r< z 1 z >1 I3 r = E t ū dµz. z >r heorem 6.1. Error estimates Assume A. A.4 hod, and et u be the entropy soution of 1.1. a Let U be a soution of the impicit method 3.5 and ū defined by 5.1. hen for a > 0, 0 < δ <, and 6.1 < r 1, u, ū, L1 R d C + E δ u E δ ū + I,r 1 + I,δ,r b Assume aso 4.3 hods, and et U be a soution of the expicit method 3.6 and ū defined by 5.. hen for a > 0, 0 < δ <, and < r 1, 6. u, ū, L 1 R d C + E δ u E δ ū + I,r 1 + I,δ,r + I r 3 he proof of this resut wi be given in Section 8. Coroary 6. Convergence. Under the assumptions of heorem 6.1, the soutions of the impicit method 3.5 and the expicit method 3.6 both converge to the unique entropy soution of 1.1 as, t 0. Proof. he resut foows from the error estimates of heorem 6.1 by first sending, t 0, then r 0, and finay ε, δ 0. We wi now see how heorem 6.1 aong with Lemma 5.4 can be used to produce expicit rates of convergence for our scheme in the case of fractiona measures satisfying 1.5. First we define τ 1 λ 0, 1, 6.3 σ IM λ τ = and 6.4 σ EX λ τ = { τ 1 og τ λ = 1, τ λ λ 1,, τ 1 λ 0, 3], τ λ +λ λ 3, 1 1,. heorem 6.3. Convergence rate for fractiona measures Under the assumptions of Lemma 5.4 incuding 1.5 and a CFL condition for the impicit scheme, for a λ 0,, C σλ IM for the impicit method 3.5, u, ū, L 1 R d C σλ EX for the expicit method 3.6. Note that the rate for the expicit method is worse due to the extra term I r 3 in heorem 6.1.,,

20 0 S. CIFANI AND E. R. JAKOBSEN Coroary 6.4 Expicit scheme when λ = 1. Let the assumptions of Lemma 5.4 b hod with λ = 1 and et α 1, be arbitrary. If the stronger CLF condition < 1 hods, then C t α u, ū, L1 R d C σ EX α for the expicit method 3.6. Proof. Note that the CFL condition 4.4 is satisfied and that the assumption 1.5 hods with any λ [1,. Hence the resut foows from the λ > 1 case in heorem 6.3. Proof of heorem 6.3. Let us first give the proof for the impicit method 3.5. First we note that by 1.5, z z dµz c λ z d+λ dz O r λ for a λ 0,, r 1, whie z r r< z 1 z r z dµz c λ r< z 1 z dz = z d+λ O1 if λ 0, 1, O n r if λ = 1, O r 1 λ if λ 1,. Using these estimates aong with the CFL condition 5.7 and Lemma 5.4, we find that the estimate 6.1 in heorem 6.1 takes the form + δ + r λ + + δ if λ 0, 1, u, ū, L1 R d C + δ n δ + r + n r + δ if λ = 1, + δ 1 λ + r λ + r 1 λ + λ δ if λ 1,. he concusion then foows by taking r = for a λ 0,, = δ = for λ 0, 1], whie = λ and δ = λ for λ 1,. For the expicit method 3.6 we aso need to take into account the extra I 3 -term, I3 r = σ λ t dµz, z >r } {{ } Or λ Lemma 5.4, and the sighty more restrictive CFL condition 4.4. he expression 6. in heorem 6.1 then takes the form u, ū, L 1 R d + δ + r λ C + δ 1 λ + + δ + + r λ + r 1 λ if λ 0, 1, r λ + if λ 1,. r λ + λ δ We minimize two and two terms and take the maximum minimizers, first w.r.t. ε and δ and then w.r.t. r, u, ū, L 1 R d r λ + 1 +, if λ 0, 1 r C λ r λ + r 1 λ 1 + r 1 λ λ 1+λ 1+λ + if λ 1,, r λ 1 + λ +λ if λ 0, 1, C λ + λ λ 3 λ + +λ if λ 1,.

21 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 1 he fina resut foows since λ 3 λ > λ +λ for λ 1 λ, and +λ < 1 for λ 3,. Remark 6.5. he rates can not be improved by taking a different truncation of the singuarity, i.e. repacing in the method G α, = 1 d 1 R x + z 1 R x dµz dx by z > G α, = 1 d 1 R x + z 1 R x dµz dx. z >ρ λ he reason is that the function ρ λ that minimize the error expression + δ + ρ λ λ is aways ρ λ = O! + ρ 1 λ λ + δ Remark 6.6. We beieve that the rates for the impicit schemes are optima, at east when there are noninear convection terms in the equation i.e. when f 0 in 1.1, see Section 7. But we have not found anaytica exampes confirming this, nor have we been abe to observe the above rates in preiminary, but probaby too crude, numerica tests. Maybe it is not straight forward to construct anaytica or numerica exampes confirming the optimaity of the rates. We eave it as a chaenge for peope with more experience in reaizing numerica schemes to test the optimaity numericay. 7. Convection-diffusion equations In this section we discuss how to extend the resuts estabished in the previous sections to the case f 0. Note that a the arguments needed to hande the additiona f-term are we-known. We consider the foowing numerica methods d 7.1 Uα n+1 = Uα n + t f Uα n+1, Uα+e n+1 + t L µ AU n+1 α, impicit U n+1 α U n+1 α = U n α + t = U n α + t =1 d =1 d =1 D D D, f U n α, U n α+e + t L µ AU n+1 α, f U n α, U n α+e + t L µ AU n α, exp-imp expicit where i D U α = 1 U α U α e and {e } is the standard basis of R d, and ii f = f 1,..., f d is a consistent i.e. fu, u = fu, Lipschitz continuous numerica fux which is non-decreasing w.r.t. the first variabe and nonincreasing w.r.t. the second one. Remark 7.1. Some exampes of numerica fuxes f satisfying ii are the we-known Lax-Friedrichs fux, the Godunov fux, and the Engquist-Osher fux, cf. e.g. [6]. 7.4 For the schemes 7. and 7.3, we aso need the CFL conditions t d L F + cl t A σ µ < 1 and dl t F < 1 respectivey compare with 4.3, where σ µ is defined in 4.1 and L F is the Lipschitz constant of f. hen the a the a priori estimates and other resuts of Section

22 S. CIFANI AND E. R. JAKOBSEN 5 continue to hod for the new schemes, and we sti have compactness via Komogorov s theorem. he modifications needed to identify the any imit as the unique entropy soution of 1.1 are standard and can be found e.g. in Chapter 3 in [], and hence the convergence of the methods foows. We wi now give the statement of the resut of heorem 6.1 that is vaid for the current setting where f 0. o do so we reuse the quantities I,r 1 and I3 r of section 6, but redefine I,δ,r as foows I,δ,r = t δ r< z 1 z dµz + dµz. z >1 heorem 7.. Error estimates Assume A.1 A.4 hod, and et u be the entropy soution of 1.1. a Let U be a soution of 7.1 or 7. and ū defined by 5.1. For 7. we aso need the second CLF condition in 7.4. hen for a > 0, 0 < δ <, and < r 1, u, ū, L1 R d C + E δ u E δ ū + I,r 1 + I,δ,r. b Assume aso that the first CFL condition in 7.4 hods, and et U be a soution of 7.3 and ū defined by 5.. hen for a > 0, 0 < δ <, and < r 1, u, ū, L 1 R d C + E δ u E δ ū + I,r 1 + I,δ,r + I r 3 he proof is essentiay equa to the proof of heorem 6.1 augmented by standard Kuznetsov type computations to hande the f-term, cf. e.g. [, Exampe 3.14]. We skip it. Remark 7.3. It is easy to see that the contribution to the error from the discretization of the f-term is aways ess or of the same order as the contributions of the other terms. In particuar, for fractiona measures 1.5, we immediatey get that the schemes satisfy the error estimate of heorem 6.3 with moduus σλ IM for 7.1 and 7. and moduus σλ EX for he proof of heorem 6.1 Proof of heorem 6.1 for the impicit method We use Lemma. to compare the soution of the scheme to the exact soution. In the resuting inequaity, we introduce the scheme via the time derivative and the initia/fina terms. o do this, we use integration by parts on each interva t n, t n+1 and summation by parts to get discrete time derivatives on ū so that we can use the ce entropy inequaity 4.5. We get that remember the definition of ū ηūx, t, uy, s t ϕ,δ x, y, t, sdw + initia and fina terms = N 1 n=0 α Z d ηuα n+1, uy, s ηuα n, uy, s ϕ,δ x, y, t n+1, s dx dyds. Let ϕ,δ = ϕ,δ x, y, t, s be the function which for each y, s is defined by ϕ n α = 1 d ϕ,δ x, y, t n, s dx for x, t t n 1, t n ],

23 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 3 and use above equation aong with the ce entropy inequaity 4.5 and Lemma 3.1 to write the inequaity of Lemma. in the foowing way u, ū, L1 R d C + + E δ u E δ v + ηaūx, t, Auy, s L µ r [ϕ,δ x,, t, s]y dw Q } {{} H 1 + ηaūx, t, Auy, s L µ r [ ϕ,δ, y, t, s]x dw Q } {{} H + η ūx, t, uy, s L µ,r [Aū, t]x ϕ,δ ϕ,δ x, y, t, s dw Q } {{} H 3 + ηaūx, t, Auy, s γ µ,r D ϕ,δ x ϕ,δ x, y, t, s dw. Q } {{} H 4 Here we have aso used the notation L[φ]x = L r [φ]x + L r [φ]x + γ µ,r D φx where L is defined in 3.3, Lr = L r for r, and L r [φ]x = φx + z φx 1 z <1 z D φx dµz. < z <r Note that the discrete operator D = D, see 3. aways acts on the x-variabe the variabe of ū. o compete the proof we need to estimate H 1,..., H 4.. Estimates of H 1 and H. By ayor s formua with integra remainder, integration by parts, and Fubini see e.g. Lemma B.1 in [] for more detais, 1 H 1 1 τ D y ηaūx, t, Auy, s z r 0 ω δ t s Dy Ω x y + τz z dτ dµz dw }{{} = D xω x y+τz 1 L A u, s BV R d ds D x Ω x dx 0 R d C L A u 0 BV R d 1 z dµz. z r R ω ε t dt z dµz z r Here we aso used heorem.1 and the standard estimate R D d x Ω x dx = O 1 ε. We find a simiar estimate for H via a reguarization procedure and the argument for H 1 above. Let ϕ,δ ϱ be a moification in the x-variabe of ϕ,δ, i.e. ϕ,δ ϱ = ϕ,δ x Ω ρ where the convoution is in x ony. hen ϕ,δ ϱ is smooth in x, and ϕ,δ ϱ, y, t, s BV R d ϕ,δ, y, t, s BV R d ϕ,δ, y, t, s BV R d = O 1, where the first inequaity hods for a ϱ sma enough cf. e.g. [30, heorem 5.3.1], whie the second one is obvious. Let us ca H ϱ = ηaūx, t, Auy, s L µ r [ ϕ,δ ϱ, y, t, s]x dw.

24 4 S. CIFANI AND E. R. JAKOBSEN First note that im ϱ 0 H ϱ = H by the dominated convergence theorem since we are integrating away from the singuarity and ϕ,δ ϱ, y, t, s ϕ,δ, y, t, s pointwise. hen, since ϕ,δ ϱ, y, t, s is smooth, we repeat the argument used for H 1 and obtain H ϱ C L A u 0 BV Rd ϕ,δ ϱ BV Rd z dµz. < z r Since ϕ,δ ϱ BV Rd = Oε 1, we can take the imit ϱ 0 and get H C L A u 0 BV R d 1 z dµz. z r 3. Estimate of H 3. By the definition of ϕ,δ and properties of moifiers, a standard argument shows that ϕ,δ x, y, t, s ϕ,δ x, y, t, s dy ds d Ω ε BV ω δ L 1 + d Ω ε L 1 ω δ BV t O + t. δ Simiar estimates are given in e.g. [13]. his estimate aong with severa appications of Fubini s theorem then show that for a < r 1, H 3 L µ,r [Aū, t]x ϕ,δ x, y, t, s ϕ,δ x, y, t, s dy ds dx dt c L A C L A + t ūx + z, t ūx, t dµz dx dt δ r< z 1 + ūx + z, t ūx, t dµz dx dt z >1 + u 0 BV z dµz + u 0 L 1 δ r< z 1 4. Estimate of H 4. Let 0,..., d and write H 4, = γ µ,r N 1 α Z d n=0 tn+1 ηauα n, Auy, s t n z >1 dµz D ϕ,δ x, y, t, s dw } {{ } H4, 1 γ µ,r N 1 tn+1 ηauα n, Auy, s t n x ϕ,δ x, y, t, s dw. α Z d n=0 }{{} H4, Since t n+1 ϕ,δ x, y, t, s dxdt = t n+1 t n ϕ,δ x, y, t n+1, s dxdt by definition, we can use summation by parts to find that t n H4, 1 = N 1 α Z d n=0 tn+1 D ηauα n, Auy, s ϕ,δ x, y, t n+1, s dw. t n.

25 NUMERICAL MEHODS FOR CONVECION-DIFFUSION EQUAIONS 5 Integration in the x -direction foowed by summation by parts eads to N 1 H4, = D ηauα n, Auy, s α Z d n=0 tn+1 ϕ,δ x x =x α, y, t, s dx 1... dx 1 dx dx d dt dy ds. t n Here we first integrated x ϕ,δ, y, t, s aong the interva x α, x α+1 to obtain the difference ϕ,δ x x =x α+1, y, t, s ϕ,δ x x =x α, y, t, s, and then we used summation by parts to move this difference onto ηauα n, Auy, s. Note that x x =x α = x 1,..., x 1, x α, x +1,..., x d, and that x = x α is fixed here whie the other variabes x j, j vary. By the above computations, the inequaity D ηauα n, Auy, s D AUα n i.e. a k b k a b, and Fubini, we find that H 4, = γ µ,r tn+1 t n γ µ,r α Z d n=0 tn+1 t n N 1 α Z d n=0 D ηau n α, Auy, s ϕ,δ x x =x α, y, t, s ϕ,δ x α, y, t n+1, s N 1 D AU n α dx dt dy ds ϕ,δ x x =x α, y, t, s ϕ,δ x α, y, t n+1, s dw. Since φ ε,δ x, y, t, s = Ω ε x yω δ t s and x, t t n, t n+1 ], we find as in part 3 that C ϕ,δ x x =x α, y, t, s ϕ,δ x α, y, t n+1, s dyds Ω ε BV ω δ L 1 + Ω ε L 1 ω δ BV t Summing over we then find that = O ε + t. δ t H 4 d C γ µ,r δ + N 1 D AUα n t d, n=0 α Z d and since α Z d D AU n α d = Aū, t n BV L A u 0 BV, we concude that H 4 C L A + t z dµz. δ r< z 1 In view of part 1-4 the proof is now compete. Proof of heorem 6.1 for the expicit method 3.6. We argue as in the beginning of the proof for the impicit method, repacing the impicit ce entropy inequaity

26 6 S. CIFANI AND E. R. JAKOBSEN by the expicit one 4.6, and find that u, ū, L1 R d C + + E δ u E δ v + ηaūx, t, Auy, s L µ r [ϕ,δ x,, t, s]y dw Q + ηaūx, t, Auy, s L µ r [ ϕ,δ, y, t, s]x dw Q + η ūx, t + t, uy, s L µ,r [Aū, t]x ϕ,δ x, y, t, s dw Q η ūx, t, uy, s L µ,r [Aū, t]x ϕ,δ x, y, t, s dw Q + ηaūx, t, Auy, s γ µ,r D ϕ,δ x ϕ,δ x, y, t, s dw. he difference with the previous proof is the interpoation 5., and more importanty, the new L µ,r -terms. Note that by a change of variabes, η ūx, t + t, uy, s L µ,r [Aū, t]x ϕ,δ x, y, t, s dw Q = η ūx, t, uy, s L µ,r [Aū, t t]x ϕ,δ x, y, t, s dw Q t, + t + η ūx, t, uy, sl µ,r [Aū, t t]x ϕ,δ x, y, t, s dw, R d where Q a,b = R d a, b. he ast term on the right can be estimated by t φ ε,δ L 1 Aū BV z dµz + Aū L 1 dµz r< z <1 z >1 = O t z dµz. r< z <1 By simiar computations, we can write the L µ,r -terms in the above inequaity as η ūx, t, uy, s L µ,r [Aū, t t Aū, t]x ϕ,δ x, y, t, s dw Q t, }{{} I + η ūx, t, uy, s L µ,r [Aū, t]x ϕ,δ ϕ,δ x, y, t, s dw Q + O t z dµz. r< z <1 Here we estimate the first term using the time reguarity of ū, I L µ,r [Aū, t t Aū, t]x ϕ,δ x, y, t, s dw Q } {{} c L A ū, t t ū, t L1 R d dt 0 C E t ū z >r dµz, =O1 dµz z >r where E t ū is defined in.. Now a the remaining terms can be estimated as in the proof for the impicit method 3.5, so the proof is compete.

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