SEMI-LAGRANGIAN SCHEMES FOR LINEAR AND FULLY NON-LINEAR DIFFUSION EQUATIONS

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1 SEMI-LAGRANGIAN SCHEMES FOR LINEAR AND FULLY NON-LINEAR DIFFUSION EQUATIONS KRISTIAN DEBRABANT AND ESPEN R. JAKOBSEN Abstract. For lnear and fully non-lnear dffuson equatons of Bellman-Isaacs type, we ntroduce a class of approxmaton schemes based on dfferencng and nterpolaton. As opposed to classcal numercal methods, these schemes wor for general dffusons wth coeffcent matrces that may be non-dagonal domnant and arbtrarly degenerate. In general such schemes have to have a wde stencl. Besdes provdng a unfyng framewor for several nown frst order accurate schemes, our class of schemes ncludes new frst and hgher order versons. The methods are easy to mplement and more effcent than some other nown schemes. We prove consstency and stablty of the methods, and for the monotone frst order methods, we prove convergence n the general case and robust error estmates n the convex case. The methods are extensvely tested.. Introducton In ths paper we ntroduce and analyze a class of approxmaton schemes for fully non-lnear dffuson equatons of Bellman-Isaacs type, (.) u t nf sup L α,β [u](t, x) + c α,β (t, x)u + f α,β (t, x) = 0 n Q T, α A β B (.) u(0, x) = g(x) n R N, where Q T := (0, T ] R N, A and B are complete metrc spaces, and L α,β [u](t, x) = tr[a α,β (t, x)d u(t, x)] + b α,β (t, x)du(t, x). The coeffcents a α,β = σα,β σ α,β, b α,β, c α,β, f α,β and the ntal data g tae values respectvely n S N, the space of N N symmetrc matrces, R N, R, R, and R. We wll only assume that a α,β s postve sem-defnte, thus the equaton s allowed to degenerate and hence not have smooth solutons n general. Under sutable assumptons (see Secton ), the ntal value problem (.)-(.) has a unque, bounded, Hölder contnuous, vscosty soluton u. Ths functon s the upper or lower value of a stochastc dfferental game, or, f A or B s a sngleton, the value functon of a fnte horzon, optmal stochastc control problem [7]. 00 Mathematcs Subject Classfcaton. 65M, 65M5, 65M06, 35K0, 35K55, 35K65, 49L5, 49L0. Key words and phrases. Monotone approxmaton schemes, dfference-nterpolaton methods, stablty, convergence, error bound, degenerate parabolc equatons, Hamlton-Jacob-Bellman equatons, vscosty soluton. Jaobsen was supported by the Research Councl of Norway through the project Integro-PDEs: Numercal methods, Analyss, and Applcatons to Fnance.

2 DEBRABANT AND JAKOBSEN We ntroduce a famly of schemes that we call Sem-Lagrangan (SL) schemes. They are a type dfference-nterpolaton schemes and arse as tme-dscretzatons of the followng sem-dscrete equaton u t nf α A sup β B L α,β [I x u](t, x) + c α,β (t, x)u + f α,β (t, x) = 0 n X x (0, T ), where L α,β s a monotone dfference approxmaton of L α,β and I x s an nterpolaton operator on the spatal grd X x. For more detals see Secton 3. Typcally these schemes are frst order accurate wde-stencl schemes, and f the matrx a α,β s bad enough, the stencl has to eep ncreasng as the grd s refned to have convergence. They nclude as specal cases monotone schemes from [6, 0, 7, ], new versons of these schemes, and a new non-monotone spatally second order accurate compact stencl scheme. There are three man advantages of these schemes: () they are easy to understand and mplement, () they are faster than some alternatve methods, and () they are consstent and, f I x s monotone, monotone for every postve sem-defnte dffuson matrx a α,β = σα,β σ α,β. The last pont s mportant because monotone methods are nown to converge to the correct soluton [3], whle non-monotone methods need not converge [3] or can even converge to false solutons [5]. Classcal fnte dfference approxmatons (FDMs) of (.) are not monotone (of postve type) unless the matrx a α,β satsfes addtonal assumptons le e. g. beng dagonally domnant [9]. More general assumptons are gven n e. g. [6, 4] but at the cost of ncreased stencl length. In fact, Dong and Krylov [4] proved that no fxed stencl FDM can approxmate equatons wth a second dervatve term nvolvng a general postve sem-defnte matrx functon a α,β. Note that ths type of result has been nown for a long tme, see e. g. [, ]. Some very smple examples of such bad matrces are gven by ( x x ) ( ) x x x x n [0, ] α αβ, αβ β for A = B = [0, ], I (Du)(Du) Du, and these types of matrces appear n Fnance, Stochastc Control Theory, and Mean Curvature Moton. The thrd example leads to quas-lnear equatons and wll not be consdered here, we refer nstead to []. To obtan convergent or monotone methods for problems nvolvng non-dagonally domnant matrces, we now of two strateges: () The classcal method of rotatng the coordnate system locally to obtan dagonally domnant matrces a α,β, see e. g. Secton 5.4 n [9], and () the use of wde stencl methods. The two solutons seem to be somewhat related, but at least the defnng deas and mplementaton are dfferent. Both ways lead to methods that have reduced order compared to standard schemes for dagonally domnant problems, but the frst strategy seems much more dffcult to mplement. We menton that the schemes of [] for statonary Bellman equatons have much n common wth the schemes n ths paper. However the two types of schemes and ther dervaton and error analyss are dfferent n general. Other related wde-stencl schemes are the method of [6], whch s not of SL type, and varous SL le schemes for other types of equatons, e. g. the Mean Curvature Moton equaton [], Monge- Ampère equatons [4], and non-local Bellman equatons [8]. The termnology SL schemes s already used for schemes for transport equatons, conservaton laws,

3 SL SCHEMES 3 and frst order Hamlton-Jacob equatons. In the Hamlton-Jacob settng, these schemes go bac to the 983 paper [9] of Capuzzo-Dolcetta. The rest of ths paper s organzed as follows. In the next secton we explan the notaton and state a well-posedness and regularty result for (.) (.). The SL schemes are motvated and defned n an abstract settng n Secton 3, and n Secton 4 we prove that they are consstent, L -stable, and, f I x s monotone, also monotone and convergent. We provde several examples of SL schemes n Secton 5, ncludng the lnear nterpolaton SL (LISL) scheme. Ths s the basc example of ths paper, a frst order accurate wde stencl scheme that can be defned on unstructured grds. Hgher order nterpolaton s not monotone. But for essentally monotone solutons, we use thrd order monotoncty preservng cubc Hermte nterpolaton [7, 5] to obtan new schemes called monotoncty preservng cubc nterpolaton SL (MPCSL) schemes n Secton 5.3. These compact stencl schemes are second order accurate n space and frst or second order accurate n tme. In Secton 6 we dscuss varous ssues concernng the SL schemes. We compare the LISL scheme to the scheme of Bonnans-Zdan [6] and fnd that the LISL scheme s easer to understand and mplement and s faster n general. We also explan that the SL schemes can be nterpreted as collocaton methods for dervatve free equatons, and as dynamc programmng equatons of dscrete stochastc dfferental games or optmal control problems. In Sectons 7, 8 and Appendx B, we derve robust error estmates for monotone schemes for convex equatons,. e. when I x s monotone and B s a sngleton n (.). They are obtaned through the regularzaton method of Krylov [8] and apply to degenerate equatons and non-smooth solutons. Fnally, n Secton 9 our methods are extensvely tested. In partcular we fnd a frst ndcaton that the LISL and MPCSL schemes yeld much faster methods to solve the fnance problem of [4].. Notaton and well-posedness In ths secton we ntroduce notaton and assumptons, and gve a well-posedness and comparson result for the ntal value problem (.) (.). We denote by the component by component orderng n R M and the orderng n the sense of postve sem-defnte matrces n S N. The symbols and denote the mnmum respectvely the maxmum. By we mean the Eucldean vector norm n any R p type space, e. g. f X R N P (an N P matrx), then X =,j X j = tr(xx ) where X s the transpose of X. If w s a bounded functon from some set Q Q nto R, R M, or R N P, we set w 0 = sup w(t, y), [w] δ = sup (t,y) Q (t,x) (s,y) w(t, x) w(s, y) ( x y + t s / ) δ, and w δ = w 0 + [w] δ for any δ (0, ]. Let C b (Q ) and C 0,δ (Q ), δ (0, ], denote respectvely the space of bounded contnuous functons on Q and the subset of C b (Q ) n whch the norm δ s fnte. Typcally Q = Q T or Q = R N, and we wll always suppress the doman Q when wrtng norms. For smplcty, we wll use the followng assumptons on the data of (.) (.): (A) For any α A and β B, a α,β = σα,β σ α,β for some N P matrx σ α,β. Moreover, there s a constant K ndependent of α, β such that g + σ α,β + b α,β + c α,β + f α,β K.

4 4 DEBRABANT AND JAKOBSEN These assumptons are standard and ensure comparson and well-posedness of (.) (.) n the class of bounded x-lpschtz functons. Proposton.. Assume that (A) holds. Then there exst a unque soluton u of (.) (.) and a constant C only dependng on T and K from (A) such that u C. Furthermore, f u and u are sub- and supersolutons of (.) satsfyng u (0, ) u (0, ), then u u. The proof s standard. Assumpton (A) can be relaxed n many ways, e. g. usng weghted norms, Hölder or unform contnuty, etc. But n dong so, solutons can become unbounded and less smooth, and the analyss becomes harder and less transparent. Therefore we wll not consder such extensons n ths paper. By solutons n ths paper we always mean vscosty solutons, see e. g. [0, 7]. 3. Defnton of SL schemes In ths secton we propose a class of approxmaton schemes for (.) (.) whch we call Sem-Lagrangan or SL schemes. Ths class ncludes (parabolc versons of) the control schemes of Menald [0] and Camll and Falcone [7] and the monotone schemes of Crandall and Lons []. It also ncludes SL schemes for frst order Bellman equatons [9, 6], and some new versons as dscussed n Secton 5. For a motvaton for the name, see Remar 6.. For the tme dscretzaton we use a new generalzed md-pont rule that ncludes explct, mplct, and a second order Cran-Ncolson le approxmatons. Snce the equaton s non-smooth n general, the usual way of defnng a Cran-Ncolson scheme [] only gves a frst order accurate scheme n tme. The schemes are defned on a possbly unstructured famly of grds G t, x, G = G t, x = (t n, x ) n N0, N = t n n N0 X x, for t, x > 0. Here 0 = t 0 < t < < t n < t n+ satsfy max n t n t where t n = t n t n, and X x = x N s the set of vertces or nodes for a non-degenerate polyhedral subdvson T x = Tj x j N of R N,. e., the polyhedrons Tj x satsfy nt(tj x T x ) =, Tj x = R N, j j N ρ x supdam B T x supdam T j j x x j N j N for some ρ (0, ), where dam s the dameter of the set and B T x s the greatest j ball contaned n Tj x. To motvate the numercal schemes, we wrte σ = (σ, σ,..., σ m,..., σ P ) where σ m s the m-th column of σ and observe that for > 0 and smooth functons φ, tr[σσ D φ(x)] = P tr[σ m σ md φ(x)] = m= P m= φ(x + σ m ) φ(x) + φ(x σ m ) + O( ),

5 SL SCHEMES 5 bdφ(x) = φ(x + b) φ(x) + O( ) = φ(x + b) φ(x) + φ(x + b) + O( ). These approxmatons are monotone (of postve type) and the errors are bounded by 48 P σ 4 0 D 4 φ 0 and b 0 D φ 0 respectvely. To relate these approxmatons to a grd G, we replace φ by ts nterpolant I x φ on that grd and obtan tr[σσ D φ(x)] bdφ(x) P m= (I x φ)(x + σ m ) (I x φ)(x) + (I x φ)(x σ m ), (I x φ)(x + b) (I x φ)(x) + (I x φ)(x + b). If the nterpolaton s monotone (postve) then the full dscretzaton s stll monotone and represents a typcal example of the dscretzatons we consder below. To construct the general scheme, we generalze the above constructon. Consder general fnte dfference approxmatons of the dfferental operator L α,β [φ] n (.) defned as (3.) L α,β [φ](t, x) := M = φ(t, x + y α,β,+, (t, x)) φ(t, x) + φ(t, x + y α,β,, (t, x)), for > 0 and some M, where for all smooth functons φ, (3.) L α,β [φ] L α,β [φ] C( Dφ D 4 φ 0 ). Ths approxmaton and nterpolaton yeld a sem-dscrete approxmaton of (.), U t nf sup L α,β α A [I x U](t, x) + c α,β (t, x)u + f α,β (t, x) = 0 n (0, T ) X x, β B and the fnal scheme can then be found after dscretzng n tme usng a parameter θ [0, ], (3.3) δ tn U n = nf sup L α,β α A [I x U θ,n ] n +θ + c α,β,n +θ U θ,n + f α,β,n +θ β B n G, where U n δ t φ(t, x) = As ntal condtons we tae = U(t n, x ), f α,β,n +θ = f α,β (t n +θ t n, x ),... for (t n, x ) G, φ(t, x) φ(t t, x), and φ θ,n = ( θ)φ n + θφ n, t I x φ θ,n = ( θ)i x φ n + θi x φ n. (3.4) U 0 = g(x ) n X x. Remar 3.. For the choces θ = 0,, and / the tme dscretzaton corresponds to respectvely explct Euler, mplct Euler and mdpont rule. For θ = /, the full scheme can be seen as generalzed Cran-Ncolson type dscretzaton.

6 6 DEBRABANT AND JAKOBSEN 4. Analyss of SL schemes In ths secton we prove that the SL scheme (3.3) s consstent and L -stable, and n the case when the nterpolaton (and hence the scheme) s monotone, we present exstence, unqueness, and convergence results for the schemes. Error estmates are gven n Secton 7 for the monotone convex case. For the approxmaton L α,β and nterpolaton I x we assume (Y) (I) (I) (I ) M = M = M = M = [y α,β,+, [y α,β,+, + y α,β,, ] = b α,β + O( 4 ), y α,β,+, [y α,β,+,,j y α,β,+,,j y α,β,+,,j 3 [y α,β,+,,j y α,β,+,,j y α,β,+,,j 3 y α,β,+,,j 4 + y α,β,, y α,β,, ] = σ α,β σ α,β + O( 4 ), + y α,β,,,j y α,β,,,j y α,β,,,j 3 ] = O( 4 ), + y α,β,,,j y α,β,,,j y α,β,,,j 3 y α,β,,,j 4 ] = O( 4 ), for all j, j, j 3, j 4 =,,..., N ndcatng components of the y-vectors. There are K 0, r N such that for all smooth functons φ, (I x φ) φ 0 K D r φ 0 x r. There s a set of non-negatve functons w j (x) j such that (I x φ)(x) = j and for all, j N, φ(x j )w j (x), w j (x) 0, w (x j ) = δ j, and w (x). Assumpton (I) holds, but w j = w φ,j s allowed to depend on φ. Under assumpton (Y), a Taylor expanson shows that L α,β s a second order consstent approxmaton satsfyng (3.). If we assume also (I), t then follows that L α,β [I x φ] s a consstent approxmaton of L α,β [φ] f xr 0. Indeed L α,β [I x φ] L α,β [φ] L α,β [I x φ] L α,β where L α,β [φ] L α,β [φ] s estmated n (3.), and by (I), L α,β [I x φ] L α,β [φ] C D r x r φ 0. [φ] + L α,β [φ] L α,β [φ], Remar 4.. Assumpton (Y) s smlar to the local consstency condtons used n [9]. The O( 4 ) terms nsure that the method s second order accurate as 0. Convergence wll stll be acheved f we relax O( 4 ) to o( ) as 0. Remar 4.. An nterpolaton satsfyng (I ) s sad to be postve and preserves postvty of the data. Such an nterpolaton does not use (exact) dervatves to reconstruct the functon φ, and t may be a non-monotone and non-lnear operator, as n the case of monotoncty preservng cubc nterpolaton (see Secton 5.3). When (I) holds and w φ, = w s ndependent of φ, the nterpolaton s a lnear operator and monotone n the sense that U V mples that I x U I x V.

7 SL SCHEMES 7 The w j s form a bass and the relaton w φ,(x) follows readly from the other assumptons n (I) and (I). Examples are constant, lnear, or mult-lnear nterpolaton (. e. r n (I)) snce hgher order nterpolaton s not monotone. The scheme s sad to be of class L f t can be wrtten as sup nf B α,β,n,n U α β n,j,j U j n B α,β,n,n U n,j, U n j (4.) B α,β,n,n U n,j, U n F α,β,n j = 0 n G, where B α,β,n,m U m,,j 0 mght depend on U m. In the case of monotone nterpolaton, B α,β,n,m U m,,j are ndependent of U m, and the L-property mples monotoncty of the approxmaton scheme n the sense of Barles-Sougands [3]. In the followng, we denote by c α,β,+ the postve part of c α,β. We now show consstency and stablty of the scheme. Lemma 4. (All SL schemes). Assume (I), (I ), and (Y) hold. (a) The scheme (3.3) s consstent wth (.) wth truncaton error bounded by ( θ φ tt 0 t + C t ( φ tt 0 + φ ttt 0 + Dφ tt 0 + D ) φ tt 0 + D r x r φ 0 + ( Dφ D 4 φ 0 ) ). (b) The scheme (3.3) s of class L (see (4.) for the defnton) f the followng CFL condton holds, [ M ] (4.) ( θ) t cα,β,n +θ and θ t c α,β,n +θ for all α, β, n,. (c) If n addton (A) and (4.) hold and θ t sup α,β c α,β,+ 0, then any soluton U of (3.3) (3.4) s L -stable satsfyng U n 0 e sup α,β cα,β,+ 0t n [ g 0 + t n sup f α,β 0 ]. α,β Remar 4.3. By parabolc regularty D t, so D φ tt 0 φ ttt 0. When θ = /, the scheme (3.3) s second order accurate n tme. Proof. (a) The scheme (3.3) s consstent wth (.) wth a truncaton error bound θ φ tt 0 t + 3 φ ttt 0 t + sup L α,β [φ θ,n ] L α,β [I x φ θ,n ] 0 α,β,n L + sup α,β [φ n +θ φ θ,n ] 0 + c α,β,n +θ (φ n +θ φ θ,n ) 0 α,β,n for smooth φ. Part (a) now follows snce by (I), (3.), and smple computatons, L α,β [φ θ,n ] L α,β [I x φ θ,n ] C D r x r φ 0 + C( Dφ D 4 φ 0 ), L α,β [φ n +θ φ θ,n ] t θ( θ) sup L α,β [φ tt ] 0 α,β C t ( Dφ tt 0 + D φ tt 0 ), c α,β,n +θ (φ n +θ φ θ,n ) Cθ( θ) t φ tt 0.

8 8 DEBRABANT AND JAKOBSEN (b) Note that snce w φ,, where l α,β,n +θ φ,j, = L α,β [I x φ(t, )](t n +θ, x j ) = N M l= l α,β,n +θ [ φ,j, φ(t, x ) φ(t, x j ) ], ( w φ, xj + y α,β,+,l (t n +θ, x j ) ) ( + w φ, xj + y α,β,,l (t n +θ, x j ) ) wth lα,β,n +θ φ,j, = M. The l α,β,n +θ φ,j, s are non-negatve by (I). The coeffcents n (4.) can now be wrtten as B α,β,n,n U n,j,j = + θ t n ( M l α,β,n +θ U n,j,j c α,β,n +θ j ), B α,β,n,n U n,j,j = ( θ) t n ( M l α,β,n +θ U n,j,j c α,β,n +θ j ), B α,β,n,n U n,j, = θ t n l α,β,n +θ U n,j,, B α,β,n,n U n,j, = ( θ) t n l α,β,n +θ U n,j,, where j. These coeffcents are postve f (4.) holds. (c) Fx any ε > 0 and let j be such that Uj n U 0 ε. Assume frst that Uj n 0. By the defnton and sgn of the B-coeffcents (see part (b)), ( ) ( B α,β,n,n M ) U n,j,j U j n c α,β,+ 0 Uj n + θ t n lα,β,n +θ U n,j,j ( U n 0 ε), θ t n sup α,β ) U n 0, ( B α,β,n,n M U n,j, U n θ t n lα,β,n +θ U n,j,j j ( B α,β,n,n U n,j, U n + ( θ) t n sup c α,β,+ 0 ) U n 0. α,β By (4.) we then fnd that Uj n = Uj n ( + ( θ) t supα,β c α,β,+ 0 )[ θ t sup α,β c α,β,+ U n 0 + t n sup f α,β M ] 0 + θ t n 0 ε ] e sup α,β cα,β,+ 0t n [ g 0 + t n sup f α,β 0 + O,n (ε). α,β If Uj n < 0, a smlar argument shows that Ũ = U satsfes the same nequalty. Snce U n 0 Uj n + ε and ε s arbtrary, the result then follows. If monotone nterpolaton s used, we also prove exstence, unqueness, and convergence of the schemes. Theorem 4. (Monotone SL schemes). Assume (A), (I), (I), (Y), and (4.). (a) There exsts a unque bounded soluton U of (3.3) (3.4). (b) U converges unformly to the soluton u of (.) (.) as t,, xr 0. Proof. (a) Exstence and unqueness follow by nducton. Let t = t n and assume U n s a nown bounded functon. For ε > 0 we defne the operator T by T U n j = U n j ε (left hand sde of (4.)) for all j Z M. α,β

9 SL SCHEMES 9 Note that the fxed pont equaton U n = T U n s equvalent to equaton (3.3). By the defnton and sgn of the B-coeffcents (whch do not depend on U n n ths proof!) we see that T U n j sup α,β T Ũ n j ( ε [ ε ( + ε t n θ [ + t n θ( M l α,β,n +θ j,j ( M t n θ sup α,β l α,β,n +θ j,j ) U n Ũ n 0 c α,β,+ 0 ]) U n Ũ n 0 c α,β,n +θ j ) )] (Uj n Ũ j n ) for ε such that ε(+ tθ( M c α,β,n +θ j )) 0 and ε( t n θ sup α,β c α,β,+ 0 ) < for all j, n, α, β. Tang the supremum over all j and nterchangng the role of U and Ũ proves that T s a contracton on the Banach space of bounded functons on X x under the sup-norm. Exstence and unqueness then follows from the fxed pont theorem (for U n ) and for all of U by nducton snce U 0 = g s bounded. (b) In vew of the L -stablty of the scheme (Lemma 4. (c)), convergence of U to the soluton u of (.) (.) follows from the Barles-Sougands result n [3]. 5. Examples of SL schemes 5.. Examples of approxmatons L α,β. We present several examples of approxmatons of the term L α,β [φ] of the form L α,β [φ], ncludng prevous approxmatons that have appeared n [6, 0, 7, ] plus more computatonal effcent varants.. The approxmaton of Falcone [6] (see also [9]), b α,β Dφ I xφ(x + hb α,β ) I x φ(x), h corresponds to our L α,β f = h, y α,β,± = b α,β.. The approxmaton of Crandall-Lons [], tr[σα,β σ α,β D φ] P j= I x φ(x + σ α,β j ) I x φ(x) + I x φ(x σ α,β j ), corresponds to our L α,β f y α,β,±,j = ±σ α,β j and M = P. 3. The corrected verson of the approxmaton of Camll-Falcone [7] (see also [0]), tr[σα,β σ α,β D φ] + b α,β Dφ P I x φ(x + hσ α,β j + h P bα,β ) I x φ(x) + I x φ(x hσ α,β j + h P bα,β ), h j= corresponds to our L α,β f = h, y α,β,±,j = ±σ α,β j + P bα,β and M = P. 4. The new approxmaton obtaned by combnng approxmatons and, tr[σα,β σ α,β D φ] + b α,β Dφ I xφ(x + b α,β ) I x φ(x) + P j= I x φ(x + σ α,β j ) I x φ(x) + I x φ(x σ α,β j ),

10 0 DEBRABANT AND JAKOBSEN corresponds to our L α,β f y α,β,±,j = ±σ α,β j for j P, y α,β,±,p + = b α,β and M = P The new, more effcent verson of approxmaton 3, P tr[σα,β σ α,β D φ]+b α,β Dφ + I xφ(x + σ α,β P corresponds to our L α,β and M = P. j= I x φ(x + σ α,β j ) I x φ(x) + I x φ(x σ α,β j ) + b α,β ) I x φ(x) + I x φ(x σ α,β P + b α,β ), f y α,β,±,j = ±σ α,β j for j < P, y α,β,±,p = ±σ α,β P + b α,β Approxmaton 5 s always more effcent than 3 n the sense that t requres fewer arthmetc operatons. The most effcent of approxmatons 3, 4, and 5, s 4 when σ α,β does not depend on α, β but b α,β does, and 5 n the other cases. 5.. Lnear nterpolaton SL scheme (LISL). To eep the scheme (3.3) monotone, lnear or mult-lnear nterpolaton s the most accurate nterpolaton one can use n general. In ths typcal case we call the full scheme (3.3) (3.4) the LISL scheme, and we wll now summarze the results of Secton 4 for ths specal case. Corollary 5.. Assume that (A) and (Y) hold. (a) The LISL scheme s monotone f the CFL condtons (4.) hold. (b) The truncaton error of the LISL scheme s O( θ t + t + + x ), so t s frst order accurate when = O( x / ) and t = O( x), or f θ =, t = O( x / ). (c) If θ t sup α,β c α,β,+ 0 and (4.) hold, then there exsts a unque bounded and L -stable soluton U of the LISL scheme convergng unformly to the soluton u of (.) (.) as t,, x 0. From ths result t follows that the scheme s at most frst order accurate, has wde and ncreasng stencl and a good CFL condton. From the truncaton error and the defnton of L α,β the stencl s wde snce the scheme s consstent only f x/ 0 as x 0 and has stencl length proportonal to l := max t,x,α,β, yα,β,, x y α,β,+, as x 0. x Here we have used that f (A) holds and σ 0, then typcally y α,β,±,. Note that f = x /, then l x /. Fnally, n the case θ the CFL condton for (3.3) s t C x when = O( x / ), and t s much less restrctve than the usual parabolc CFL condton, t = O( x ) A hgh order SL scheme for monotone solutons. In ths secton we ntroduce spatally second order accurate SL schemes (3.3) (3.4) for non-degenerate tensor product grds. These schemes are based on monotoncty preservng cubc (MPC) Hermte nterpolaton [7, 5] and wll be denoted MPCSL schemes n short. They are consstent for monotone (n coordnate drectons) solutons of the scheme, but they are not monotone.

11 SL SCHEMES The MPC nterpolaton s obtaned by a careful modfcaton of cubc Hermte nterpolaton [5], and for a functon of one varable on the nterval [x, x + ] t taes the form (5.) (I x φ)(x) = φ + (φ + φ )P (x) where ( ) ( ) 3 x x x P (x) = α x + (3 β x x x (5.) α ) ( α β ), x x where α, β are bounded coeffcents dependng on φ, φ,..., φ +3. The algorthm s descrbed n Appendx A. Multdmensonal nterpolaton operators are obtaned as tensor products of one-dmensonal nterpolaton operators,. e. by nterpolatng dmenson by dmenson. Remar 5.. Rewrtng I x φ, we fnd that (I x φ)(x) = φ w φ, (x) for w φ, (x) = ( P (x)) [x,x +)(x) + P (x) [x,x )(x) and [x,x +](x) s the ndcator functon that s n [x, x + ) and 0 otherwse. It s mmedate that w φ,(x), and w φ, 0 snce P (x ) = 0, P (x + ) =, and P s monotone n between. Lemma 5.. The above monotoncty preservng cubc nterpolaton satsfes (I ). If the nterpolated functon s strctly monotone between grd ponts, then (I) holds wth r = 4 and the method s fourth order accurate. Proof. Assumpton (I) holds by constructon, see remar 5.. The error estmate follows from [5], snce the above algorthm concdes wth the two sweep algorthm gven there when n = nterval s consdered. In [5] t s proved that ths algorthm gves thrd order accurate approxmatons to the exact dervatves and hence the cubc Hermte polynomal constructed usng ths approxmaton s fourth order accurate. By Lemma 5. and the results n Secton 4 we have the followng result: Corollary 5.3. Assume (A), (Y) hold, and that for all x (0, ), solutons U of the MPCSL scheme are such that I x U s strctly x-monotone between ponts n the x-grd X x. (a) The truncaton error of the MPCSL scheme s ) O ( θ t + t + + x4, and hence the scheme s second order accurate n space when = O( x) and frst or second order accurate n tme when θ or θ = respectvely. (b) If θ t sup α,β c α,β 0, then the soluton U s L -stable. 6. Dscusson 6.. Comparson wth the scheme of Bonnans-Zdan (BZ). In [6] (see also [5, 4]) Bonnans and Zdan suggest an alternatve approach to dscretze degenerate dffuson equatons. Ther dea s to approxmate the dffuson matrx a α,β by a ncer matrx a α,β whch admts monotone fnte dfference approxmatons. For every N they fnd a stencl

12 DEBRABANT AND JAKOBSEN S ξ = (ξ,..., ξ N ) Z N : 0 < and postve numbers a α,β,ξ such that a α,β a α,β := N max = ξ, =,..., N ξ S a α,β,ξ ξξ. Ths leads to a dffuson term that s a lnear combnaton of drectonal dervatves whch are agan approxmated by central dfference approxmatons, tr[a α,β D φ] tr[a α,β D φ] = a α,β,ξ D ξφ a α,β,ξ ξφ, ξ S ξ S where D ξ = tr[ξξt D ] = (ξ D) and ξ w(x) = ξ w(x + ξ x) w(x) + w(x ξ x). x Ths approxmaton s monotone by constructon and respects the grd. In two space dmensons, a α,β can be chosen such that a α,β a α,β = O( ) (cf. [5]), and then t s easy to see that the truncaton error s O( + x ). When b α,β 0, the BZ scheme can be obtaned from (3.3) by replacng our L α,β by the above Bonnans-Zdan dffuson approxmaton. Ths scheme shares many propertes wth the LISL scheme, t s at most frst order accurate (tae x / ), t has a smlar wde and ncreasng stencl, and t has a smlar good CFL condton t C x ( x when x / ). To understand why the stencl s wde, smply note that by defnton s the stencl length and that the scheme s consstent only f and x 0. The typcal stencl length s x /, just as t was for the LISL scheme. The man drawbac of ths method s that t s costly snce we must compute the matrx a α,β for every x, t, α, β n the grd. In the fast two dmensonal mplementaton n [5], the number of operatons for computng the coeffcents for a fxed x, t, α, β s O() and thus goes to nfnty as n bad cases. The LISL scheme s easer to understand and mplement and s faster n the sense that the computatonal cost for approxmatng the dffuson matrx for fxed x, t, α, β s ndependent of the stencl sze. Later we wll see some numercal ndcaton that the LISL scheme could be faster than the BZ scheme n some test problems. The MPCSL scheme n the typcal case when = x, s a second order accurate n space and compact stencl scheme havng the usual (not so good) CFL condtons for parabolc problems t = x. When t can be used, t s far more effcent than the other two schemes, see Secton 9. However there s no proof that the method wll converge to the correct soluton, and t s formally convergent only when the exact solutons are essentally monotone, meanng monotone at least between grd ponts. Both the BZ and LISL schemes always converge. 6.. Boundary condtons. When solvng PDEs on bounded domans, the SL (and BZ) schemes may exceed the doman f they are not modfed near the boundary. The reason s of course the wde stencl. Ths may or may not be a problem dependng on the equaton and the type of boundary condton: () For Drchlet condtons the scheme needs to be modfed near the boundary or boundary condtons must

13 SL SCHEMES 3 be extrapolated. Ths may result n a loss of accuracy or monotoncty near the boundary. () Homogeneous Neumann condtons can be mplemented exactly by extendng n the normal drecton the values of the soluton on the boundary to the exteror. () If the boundary has no regular ponts, no boundary condtons can be mposed. In ths case the SL schemes wll not leave the doman f the normal dffuson tends to zero fast enough when the boundary s approached. Typcal examples are equatons of Blac-Scholes type Interpretaton as a collocaton method. In the case the functons w φ, n (I) do not depend on φ (and form a bass), the scheme (3.3) (3.4) can then be nterpreted as a collocaton method for a dervatve free equaton, ths s essentally the approach of Falcone et al. [6, 7]. The dea s that f W x (Q T ) = u : u s a functon on Q T satsfyng u I x u n Q T denotes the nterpolant space assocated to the nterpolaton I x, equaton (3.3) can be stated n the followng equvalent way: Fnd U W x (Q T ) solvng (6.) δ tn U n = nf sup L α,β α A [U θ,n ] n +θ + c α,β,n +θ U θ,n + f α,β,n +θ n G. β B In general W x can be any space of approxmatons whch s nterpolatng on the grd X x, e. g. a space of splnes, but we do not consder ths generalty here Stochastc game/control nterpretaton. The scheme (3.3) (3.4) can be nterpreted as the dynamcal programmng equaton of a dscrete stochastc dfferental game. We wll explan ths n the less techncal case when B s a sngleton and the game smplfes to an optmal stochastc control problem. Assume that (A) holds, and for smplcty, that c α (t, x) 0 and the other coeffcents are ndependent of t. Then t s well-nown (cf. [7]) that the (vscosty) soluton u of (.) (.) s the value functon of the stochastc control problem: [ T ] (6.) u(t t, x) = mn E f α(s) (X s ) ds + g(x T ), α( ) A t where A s a set of admssble A-valued controls and the dffuson process X s = satsfes the SDE X t,x,α( ) s (6.3) X t = x and dx s = σ α(s) (X s ) dw s + b α(s) ds for s > t. Ths follows from dynamcal programmng (DP), and (.) s called the DP equaton for the control problem (6.) (6.3). Smlarly, the schemes (3.3) (3.4) are DP equatons (at least n the explct case) of sutably chosen dscrete tme and space control problems approxmatng (6.) (6.3). We refer to [9] for more detals. We tae the slghtly dfferent approach explored n [9, 6, 0, 7] to show the relaton to control theory. The dea s to wrte the SL scheme n collocaton form (6.) and show that (6.) s the DP equaton of a dscrete tme contnuous space optmal control problem. We llustrate ths approach by dervng an explct scheme nvolvng L α as defned n part 4 Secton 5.. Let t 0 = 0, t,..., t M = T be dscrete tmes and consder the dscrete tme approxmaton of (6.) (6.3) gven by (6.4) ũ(t t m, x) = mn α A M E [ M =m f α ( X ) t + + g( X ] M ),

14 4 DEBRABANT AND JAKOBSEN (6.5) X m = x, Xn = X n + σ αn ( X n ) n ξ n + b αn ( X n ) n η n, n > m, where n = (P + ) t n, A M A s an approprate subset of pecewse constant controls, and ξ n = (ξ n,,..., ξ n,p ) and η n are mutually ndependent sequences of.. d. random varables satsfyng ) P ((ξ n,,..., ξ n,p, η n ) = ±e j = (P + ) ) P ((ξ n,,..., ξ n,p, η n ) = e P + = P +, f j,..., P, (e j denotes the j-th unt vector) and all other values of (ξ n,,..., ξ n,p, η n ) have probablty zero. Here we have used a wea Euler approxmaton of the SDE coupled wth a quadrature approxmaton of the ntegral. By DP [ n ũ(t t m, x) = mn E α A M =m f α ( X ) t + + ũ(t t n, X ] n ) for all n > m, and tang n = m +, s M m = T t m, s m = s m s m, m = M m, and evaluatng the expectaton usng (6.5), we see that ũ(s M m, x) = mn α A f α (x) s M m + M m P + Lᾱ M m [ũ](s M m, x) + ũ(s M m, x), where L α s as n Secton 5. part 4. If we subtract ũ(s M m, x) from both sdes and dvde by s M m = M m P +, we fnd (6.) wth θ = 0. In [7], a smlar argument s gven n the statonary case for schemes nvolvng the L α of part 3 Secton 5.. In fact t s possble to dentfy all Lα s appearng n Secton 5. wth DP equatons of sutably chosen dscrete tme contnuous space control problems. However assumpton (Y) s not strong enough for ths approach to wor for the general L α defned n Sectons 3 and 4. Remar 6.. A DP approach naturally leads to explct methods for tme dependent PDEs. But mplct methods can be derved from a trc: Dscretze the PDE n tme by bacward Euler to fnd a (sequence of) statonary PDEs and use the DP approach on each statonary PDE. Ths leads to an mplct teraton scheme snce the DP equatons of statonary problems are always mplct. Remar 6.. By the defnton of L α and (Y), x + yα,±, can be seen as a short tme approxmaton of (6.3). Hence the scheme (3.3) tracs partcle paths approxmately. In vew of the dscusson above we mght say that the scheme follows partcles n the mean because of the expectaton. For frst order PDEs, schemes defned n ths way are called SL schemes by e. g. Falcone. Moreover, n ths case our schemes wll concde wth the SL schemes of Falcone [6] n the explct case. Ths explans why we choose to call these schemes SL schemes also n the general case. 7. Error estmates n the monotone convex case We derve error bounds when I x s monotone and B s a sngleton and hence (.) s convex. It s not nown how to prove such results n the general case. In the followng we do not ndcate the trval β dependence any more and we tae

15 SL SCHEMES 5 a unform tme-grd, G = t 0,,..., N T X x, for smplcty. Let Q t,t := t 0,,..., N T R N and consder the ntermedate equaton (7.) δ t V n (x) = nf α A L α [V θ,n ](t, x) + c α (t, x)v θ,n (x) + f α (t, x) for n =,, 3,..., wth ntal condton (7.) V (0, x) = g(x) n R N. t=t n +θ n R N Lemma 7.. Assume that (I) and the CFL condton (4.) hold and that sup n V n C V. If V solves (7.) (7.) and U solves (3.3) (3.4), then U V C x n G. Proof. Let W = U V and subtract the equaton for V from the one for U to fnd W n W n + t sup L α [I x W θ,n ] n +θ + c α,n +θ W θ,n α A + L α [I x V θ,n V θ,n ] n +θ n G. Let C c = max α c α,+ 0. If W n 0, we rearrange usng I x V n V n (I) 0 = w j ( ) ( Vj n V n ( ) ) 0 w j ( ) Vj n V n ( ) to see that j x V n w j ( ) ( M )) + θ t( C c ( W n + t sup θ α A ( + ( θ) j W n 0 (I) j = x V n L α [I x W n ] n +θ L α [I x W n ] n +θ + t sup V n x n n G. + c α,n +θ + M W n ) ) W n By the CFL condton (4.), the coeffcents of the above nequalty are all nonnegatve. Hence snce W n W n 0 := sup W n, we may replace W n by W n 0 on the rght hand sde. Moreover, snce I x W n 0 = W n 0 and L α [ W n 0 ] = 0, the upper bound on the rght hand sde then reduces to ( + t( θ)c c ) W n 0 + θ t M W n 0 + C V t x. If W n < 0, then the same bound also holds for W n, and hence (+ tθ( M C c)) W n 0 (+ t( θ)c c ) W n 0 +θ t M W n 0 +C V t x t. Snce W 0 0 n X x, an teraton then reveals that W n 0 C V t x n m=0 ( + t( θ)cc tθc c ) m tn x 4C V e Cctn 0

16 6 DEBRABANT AND JAKOBSEN when t s small enough, whch mples the lemma. Next we estmate V u, where u solves (.) (.), by the regularzaton method of Krylov [8]. To do that we need a contnuty and contnuous dependence result for the scheme that reles on the followng addtonal (covarance-type) assumptons: Whenever two sets of data σ, b and σ, b are gven, the correspondng approxmatons L α, yα,±, and L α, ỹα,±, n (3.) satsfy (Y) M = M = [y α,+, + y α,, ] [ỹα,+, + ỹ α,, ] (b α b α ), [y α,+, yα,+, + y α,, yα,, ] + [ỹ α,+, ỹα,+, + ỹ α,, ỹα,, ] [y α,+, ỹα,+, + ỹ α,+, yα,+, + y α,, ỹα,, + ỹ α,, yα,, ] (σ α σ α )(σ α σ α ) + 4 (b α b α )(b α b α ), when σ, b, y ± are evaluated at (t, x) and σ, b, ỹ ± are evaluated at (t, y) for all t, x, y. In Secton 8 we wll prove the followng error estmate. Theorem 7.. Assume that B s a sngleton, that (A), (Y), (Y), and the CFL condtons (4.) hold, and that (0, ) and t ( 0 ). If u and V are bounded solutons of (.) (.) and (7.) (7.), then V u C( θ t /4 + t /3 + / ) n Q t,t. It also follows from the regularty results n Secton 8 (see Proposton 8.4) that V n C T, so by Lemma 7. and Theorem 7. we have the followng result. Corollary 7.3 (Error Bound). Under (I), (I), and the assumptons of Theorem 7., f u solves (.) (.) and U solves (3.3) (3.4), then u U u V + V U C( θ t /4 + t /3 + / + x ) n G. Ths error bound apples to the LISL schemes, and t also holds for unstructured grds. For more regular solutons t s possble to obtan better error estmates, but general and optmal results are not avalable. The best estmate n our case s O( x /5 ) whch s acheved when = O( x /5 ) and t = O( ). Note that the CFL condtons (4.) already mply that t = O( ) f θ <. Also note that the above bound does not show convergence when s optmal for the LISL scheme ( = O( x / )). Remar 7.. These results are consstent wth results for specal LISL type schemes for statonary Bellman equatons. In fact f all coeffcents are ndependent of tme and c α (x) < c < 0, then by combnng the results of [7] and [], exactly the same error estmate s obtaned for the soluton of a partcular statonary LISL scheme and the unque statonary Lpschtz soluton of (.). 8. Proof of Theorem 7. We start by an exstence and unqueness result. Lemma 8.. Assume that (A), (Y), and the CFL condtons (4.) hold. Then there exsts a unque soluton U C b (Q T, t ) of (7.) (7.).

17 SL SCHEMES 7 The proof s smlar to (but smpler than) the proof of Theorem 4. wth the modfcaton that the fxed pont s acheved n the Banach space C b (R N ) nstead of the space of bounded functons on X x. We now gve a result comparng subsolutons of (7.) to supersolutons of (8.) δ t U n (x) = nf Lα [U θ,n ](t, x) + c α (t, x)u θ,n + f α (t, x) n R N, n, α A t=t n +θ U(0, x) = g(x) n R N, where L α s the operator defned n (3.), (Y), (Y) when σα, b α are replaced by σ α, b α. Theorem 8.. Assume that (A), (Y), (4.) hold for both (7.) and (8.). If U C(Q T, t ) s a bounded above subsoluton of (7.) and Ũ C(Q T, t) a bounded below supersoluton of (8.), then for all (0, ), t ( 0 ) L0 L (see below), x, y R N, n 0,,..., N T, U(t n, x) Ũ(t n, y) R 0 (t n ) (U(0, ) Ũ(0, ))+ 0 + R 0 (t n )R (t n )(L 0 + t n L) x y [ + t n R 0 (t n ) sup (f f) ( U 0 Ũ ] 0) c c 0 α A + tn / [ ] K T sup b b 0 + σ σ 0 α A where R (t) = /( t) t/ t, K T R 0 (T )R (T )(L 0 + T L), L 0 = [g] [ g] +, L = ([c α ] [ c α ] )( U 0 Ũ 0) + [f α ] [ f α ], 0 = sup α c α,+ 0, = 8 sup[σ α ] + [b α ] +. α Remar 8.. The functon R (n t) = /( t) n satsfes δ t R (t n ) = R (t n ), R (0) =, and R (t n ) e tn when t. Ths s a ey result n ths paper, and the proof s gven n Appendx B. In the statonary case, results of ths type have been obtaned n [, 8] for smpler schemes. The result s a jont unqueness (tae ( σ, b, c, f, g) = (σ, b, c, f, g)), contnuous dependence (tae x = y), boundedness, and x-lpschtz contnuty result: Corollary 8.3. Under the assumptons of Theorem 8., f (0, ) and t ( 0 ), then any bounded soluton U C b (Q T, t ) of (7.) satsfes () U(t n, ) 0 e 0tn ( g 0 + t n sup α f α 0 ), () U(t n, x) U(t n, y) e (0+)tn (L 0 + t n L) x y, where the constants, whch are defned n Theorem 8., are ndependent of, t, x. Proof. Part () follows from Theorem 8. (wth x = y) and Remar 8. snce Ũ 0 satsfes (8.) wth ( σ α, b α, c α, f α, g α ) = (σ α, b α, c α, 0, 0). Part () follows by tang U = Ũ and x y. Now we extend the scheme (7.) to the whole space Q T. One way to do ths and to obtan contnuous n tme solutons s to pose ntal condtons on [0, t) by nterpolatng between g(x) and U( t, x) where U s the soluton of (7.) (7.).

18 8 DEBRABANT AND JAKOBSEN (8.) (8.3) δ t V (t, x) = nf L α [V θ (t, )](t θ, x) + c α (t θ, x)v θ (t, x) + f α (t θ, x) α A V (t, x) = n ( t, T ] R N, ( t ) g(x) + t t t U( t, x) n [0, t] RN. where V θ (t, x) = ( θ)v (t t, x) + θv (t, x) and t θ = t ( θ) t. From the prevous results for U the exstence, unqueness, and propertes of V easly follow. Proposton 8.4. Assume that (A), (Y), (Y), and the CFL condtons (4.) hold, and that (0, ) and t ( 0 ). (a) There exsts a unque soluton V C b (Q T ) of (8.) (8.3). (b) There s a constant C T 0 ndependent of, t, x such that () V 0 C T, () V (t, x) V (t, y) C T x y for all t [0, T ], x, y, R N, () V (s, x) V (s, x) C T s s / for all s, s [0, T ], x, R N. (c) Let V C b (Q T ) and Ṽ C b(q T ) be sub- and supersolutons of (8.) (8.3) correspondng to coeffcents (σ α, b α, c α, f α, g) and ( σ α, b α, c α, f α, g) respectvely. Then there s a constant C T 0 ndependent of, t, x such that for all t [0, T ], V (t, ) Ṽ (t, ) 0 C T ( g g 0 + t sup[( U 0 Ũ 0) c α c α 0 + f α f α 0 ] α +t / sup[ σ α σ α 0 + b α b α 0 ] α Proof. Frst note that the ntal data on [0, t] s unformly bounded and Lpschtz contnuous n x and t by constructon and Corollary 8.3. (a) Exstence of a bounded x-contnuous soluton follows from repeated use of Lemma 8. snce we have ntal condtons on [0, t]. Contnuty n tme follows from Theorem 8. (wth x = y) snce the data s t-contnuous. (b) Part () and () follow from Corollary 8.3 snce the ntal data s unformly bounded and x-lpschtz n [0, t]. To prove part () we assume s < s and let U(t, x) and Ũ(t, x) solve (8.) wth data (σα (t+s, x), b α (t+s, x), c α (t+s, x), f α (t+ s, x), V (s, x)) and (0, 0, 0, 0, V (s, x)) respectvely. Note that for t [0, T s ], Ũ(t, x) V (s, x) and U(t, x) V (t + s, x) where V s the unque soluton of (8.) (8.3). By part (c) we then get ). V (t + s, ) V (s, ) 0 = U(t, ) Ũ(t, ) 0 ) C T (0 + t sup[ f α 0 + V 0 c α 0 ] + t / sup[ σ α 0 + b α 0 ] α α for t > 0, and hence part () follows. (c) Note that by constructon of the ntal data and Theorem 8. wth x = y, the result holds for t [0, t], and then the result holds for any t > t by another applcaton of Theorem 8. wth x = y. Usng Krylov s method of shang the coeffcents [8], we wll now fnd smooth subsolutons of (8.). Frst we ntroduce the auxlary equaton

19 SL SCHEMES 9 (8.4) δ t V ε (t, x) = nf 0 s ε e ε L α [τ e V ε,θ (t, )](r + s, x + e) α A + c α (r + s, x + e)v ε,θ (t, x) + f α (r + s, x + e) (8.5) r=t θ t ε ( V ε (t, x) = t ) g(x) + t t t V ε ( t, x) n [0, t] R N, where τ e φ(t, x) = φ(t, x + e) and V ε ( t, x) s obtaned by frst solvng (8.4) for dscrete tmes t n = n t. For ths equaton to be well-defned for t ( t, T ], the data and y α,±, must be defned for t ( t ε, T + ε ]. But ths s o snce one can easly extend these functons to t [ r, T + r] for any r > 0 n such a way that (A), (Y), (Y) stll hold. Also note that (8.6) L α [τ e V ε,θ (t, )](r + s, x + e) = M V ε,θ (t, x + y α,+, (r + s, x + e)) = V ε,θ (t, x) + V ε,θ (t, x + y α,, (r + s, x + e)), and hence (8.4) s an equaton of the same type as (8.) (wth dfferent A and shfted coeffcents) satsfyng (A), (Y), (Y) whenever (8.) does. By Proposton 8.4 there s a unque soluton V ε of (8.4) (8.5) n [0, T + t + ε ] R N. Let U ε (t, x) := V ε (t + t + ε, x) and defne by convoluton, (8.7) U ε (t, x) = U ε (t s, x e)ρ ε (s, e) ds de, R N 0 where ε > 0, ρ ε (t, x) = ρ( t ε N+ ε, x ε ), and ρ C (R N+ ), ρ 0, supp ρ [0, ] x, ρ(e)de =. R N Note that U ε s well defned on the tme nterval [ t, T ]. By the next result t s the sought after smooth subsoluton of (8.). Proposton 8.5. Under the assumptons of Proposton 8.4, the functon U ε defned n (8.7) satsfes () U ε C (( t, T ) R N ), U ε C, D m n t U ε 0 Cε m n for n, m N. () If V s the soluton of (8.) (8.3), then U ε V C(ε + t / ) n Q T. () U ε s a subsoluton of (8.) n Q T. Proof. The regularty estmates n () are mmedate from propertes of convolutons and the regularty of V ε. The bound on U ε V (n [0, T ]) n () follows from Proposton 8.4 (c) and (A) whch mply V ε V 0 C(ε + t / ), and regularty of V ε along wth propertes of convolutons, U ε V ε 0 U ε U ε 0 + V ε ( + t + ε, ) V ε 0 V ε (ε + t / ). To see that U ε s a subsoluton of (8.), frst note that from the defnton of U ε and (8.4) t follows that δ t U ε (t, x) L α [τ e U ε,θ (t, )](t θ + s, x + e)

20 0 DEBRABANT AND JAKOBSEN + c α (t θ + s, x + e)u ε,θ (t, x) + f α (t θ + s, x + e) for all (t, x) [ ε, T ] R N, e, s ε, and α A. Now we change varables from (t + s, x + e) to (t, x) to fnd that δ t U ε (t s, x e) L α [τ e U ε,θ (t s, )](t θ, x) + c α (t θ, x)u ε,θ (t s, x e) + f α (t θ, x) for all (t, x) [0, T ] R N, e, s ε, and α A. Then we multply by ρ ε (s, e) and ntegrate w. r. t. (s, e). To see what the result s, note that L α [τ e U ε (t s, )](r, x) = M U ε (t s, x + y α,+, (r, x) e) = U ε (t s, x e) + U ε (t s, x + y α,, (r, x) e), and hence L α [τ e U ε (t s, )](r, x)ρ ε (s, e) ds de = L α [U ε (t, )](r, x). For the whole equaton we then have δ t U ε (t, x) L α [U θ ε(t, )](t θ, x) + c α (t θ, x)u θ ε(t, x) + f α (t θ, x) for all (t, x) Q T and α A. Snce ths nequalty holds for all α, t follows that U ε s a subsoluton of (8.) n all of Q T. We are now n a poston to prove the error estmate gven n Theorem 7.. Proof of Theorem 7.. Let U ε be defned n (8.7). By Proposton 8.5 () and Lemma 4. (a), L α [U θ ε(t, )](t θ, x) + c α (t θ, x)u θ ε(t, x) + f α (t θ, x) t U ε nf α A θ t U ε 0 t + C + ( DU ε D 4 U ε 0 ) C θ ε 3 t + ε 5 t + ε 3 n Q T. Moreover, by Proposton 8.5 (), ( t U ε t U ε 0 + t DU ε 0 + t D U ε 0 ) t g(x) = U(0, x) U ε (0, x) C(ε + t / ). It follows that there s a constant C 0 such that U ε Ce sup α cα 0t ε + t / + t ( θ ε 3 t + ε 5 t + ε 3 ) s a classcal subsoluton of (.) (.) wth tme shfted coeffcents. By contnuous dependence and the comparson prncple ( U ε Ce sup α cα 0t ε + t / + t θ ε 3 t + ε 5 t + ε 3 ) u n Q T, and hence by Proposton 8.5 (), U u = (U U ε ) + (U ε u) C ε + t / + θ ε 3 t + ε 5 t + ε 3.

21 SL SCHEMES We mnmze w. r. t. ε and fnd that C( t /4 + / ) f θ u U C( t /3 + / ) f θ = n Q T. The lower bound on u U follows wth symmetrc but much easer arguments where a smooth supersoluton of the equaton (.) s constructed. Consstency and comparson for the scheme (8.) s then used to conclude. In vew of Lemma 4., the lower bound s a drect consequence of Theorem 3. (a) n []. 9. Numercal results In the followng, we apply the LISL and MPCSL schemes to lnear and convex test problems n two space-dmensons, and hence have no dependence of β. For the LISL scheme, we choose = x and a regular trangular grd, whereas for the MPCSL scheme we choose = x and a regular rectangular grd. If not stated otherwse, we use θ = 0 (explct methods), CFL condton t =, and approxmaton 5..5 for L α,β. As error measure we wll always use the L -norm, ln e ln e and the error rates are calculated as r = ln x ln x. All calculatons are done n Matlab, on an INTEL(R) Core(TM) Duo P8700,.54Ghz Laptop. 9.. Lnear problem wth smooth soluton. Our frst problem s taen from [5] and has exact soluton u(t, x) = ( t) sn x sn x, ts coeffcents n (.) are f α (t, x) = sn x sn x [( + β )( t) ] ( t) cos x cos x sn(x + x ) cos(x + x ), c α (t, x) =0, b α (t, x) = 0, σ α (t, x) = ( ) sn(x + x ) β 0. cos(x + x ) 0 β We consder β = 0. and β = 0. Note that n the second case, the scheme consdered n [5] s not consstent. Table gves the (spatal) errors and rates obtaned at t = applyng the LISL and the MPCSL scheme, as well as the CPU tme needed. As the soluton s lnear n t, one tme step suffces. As expected for smooth solutons, n both cases we obtan order one for the LISL scheme and order two for the MPCSL scheme, and the CPU tme needed s proportonal to the number of grd ponts x. Here, we have chosen the grd ponts such that the soluton s monotone n between. If not, we would get order one for the LISL scheme but no convergence for the MPCSL scheme (see Secton 5.3). 9.. Lnear problem wth non-smooth soluton. The second problem we test has a non-smooth exact soluton u(t, x) = ( + t) sn x sn x for π < x < 0, sn x 4 for 0 < x < π n [ π, π] and coeffcents n (.) gven by f α (t, x) = sn x ( sn x + +t 4 (sn x + sn x ) ) for π < x < 0 ( sn x 4 + +t 6 (sn x + 4 sn x ) ) for 0 < x < π sn x sn x cos x +t +t 4 x cos for π < x < 0, x cos 4 for 0 < x < π

22 DEBRABANT AND JAKOBSEN (a) β = 0. x LISL MPCSL error rate tme n s error rate tme n s 3.93e- 3.79e e e-.93e e e e e e e e e-3.43e e (b) β = 0 x LISL MPCSL error rate tme n s error rate tme n s 3.93e- 3.94e e e-.98e e e e e e e e e-3.45e e Table. Results for the smooth lnear problem at t =, grd adapted to monotoncty c α (t, x) =0, b α (t, x) = 0, σ α (t, x) = ( ) sn x, sn x and we pose Drchlet boundary condtons. Ths s a monotone non-smooth problem, and we obtan order one half applyng the LISL scheme and order one applyng the MPCSL scheme,. e. reduced rates, see Table. Agan, one tme step suffces, and the CPU tme needed s thus proportonal to x. x LISL MPCSL error rate tme n s error rate tme n s 7.76e-.4e e e- 8.75e e e- 6.9e e e e e e-3 3.0e e e-3.9e e Table. Results for the non-smooth lnear problem at t = 9.3. Optmal control problems wth smooth solutons. (A) We test an example from [5] wth exact soluton u(t, x, x ) = ( 3 t) sn x sn x. The correspondng coeffcents and control set n (.) are ( ) f α = t sn x sn x + ( ) [ 3 t cos x sn x + sn x cos x

23 SL SCHEMES 3 sn(x + x ) cos(x + x ) cos x cos x ], c α = 0, b α = α, σ α = ( ) sn(x + x ), A = α R : α cos(x + x ) + α =. As σ α does not depend on α but b α does, we choose approxmaton 5..4 for L α,β and thus need only about half of the number of nterpolatons we would need f we had chosen approxmaton (B) The next test problem has exact soluton u(t, x, x ) = ( t) sn x sn x and coeffcents and control set gven by f α (t, x) = ( t) sn x sn x α α ( t) cos x cos x, c α (t, x) =0, b α (t, x, α) = 0, σ α = ( ) α, A = α R : α + α =. In both examples, due to the soluton beng lnear n t, one tme step suffces. The results at t = 0.5 are gven n Table 3, where agan the grd s adapted to monotoncty. As expected for smooth solutons, the LISL scheme yelds a numercal order of convergence of one, whereas the MPCSL scheme yelds order two. The CPU tme s now proportonal to x, reflectng that we use 4π 3 x grd ponts to dscretze the control. (a) x LISL MPCSL error rate tme n s error rate tme n s 3.93e- 3.0e e e-.6e e e e e e e e e-3.03e e (b) x LISL MPCSL error rate tme n s error rate tme n s 3.93e-.8e e e-.07e e e e e e-3.55e e e-3.34e e Table 3. Results for optmal control problems at t = 0.5, grd adapted to monotoncty α 9.4. Convergence test for a super-replcaton problem. We consder a test problem from [4] whch was used to test convergence rates for numercal approxmatons of a super-replcaton problem from fnance. The correspondng PDE s