An a posteriori error analysis based on non-intrusive spectral projections for systems of random conservation laws
AAn a posteriori error analysis based on non-intrusivespectral projections for systems of random conservationlaws
Jan Giesselmann Fabian Meyer , ˚ Christian Rohde April 30, 2019
Abstract
We present an a posteriori error analysis for one-dimensional random hyperbolic sys-tems of conservation laws. For the discretization of the random space we considerthe Non-Intrusive Spectral Projection method, the spatio-temporal discretizationuses the Runge–Kutta Discontinuous Galerkin method. We derive an a posteriorierror estimator using smooth reconstructions of the numerical solution, which com-bined with the relative entropy stability framework yields computable error boundsfor the space-stochastic discretization error. Moreover, we show that the estimatoradmits a splitting into a stochastic and deterministic part.
Key words: hyperbolic conservation laws, random pdes, a posteriori error estimates, non-intrusive spectral projection method, discontinuous galerkin method
In this contribution we study numerical schemes for spatially one-dimensional systemsof random hyperbolic conservation laws, where the uncertainty stems from random ini-tial data. The random space is discretized using the Non-Intrusive Spectral Projection(NISP) method which is based on discrete orthogonal projections, cf. [8]. The result-ing deterministic equations are discretized by a Runge–Kutta Discontinuous Galerkin(RKDG) method [2]. We reconstruct the numerical solutions based on reconstructionsfor determinstic problems suggested in [4], see also [7] for their use in Stochastic Galerkin Department of Mathematics, TU Darmstadt, Dolivostraße 15, 64293 Darmstadt, Germany. Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57,70569 Stuttgart, Germany.F.M., C.R. thank the Baden-Württemberg Stiftung for support via the project ’SEAL’. J.G. thanksthe German Research Foundation (DFG) for support of the project via DFG grant GI1131/1-1. ˚ [email protected] a r X i v : . [ m a t h . NA ] A ug chemes. Based on these reconstructions and using the relative entropy framework, cf. [3,Section 5.2], we derive an a posteriori error bound for the difference between the exactsolution of the random hyperbolic conservation law and its numerical approximation.We show that the corresponding residual admits a decomposition into three parts: Aspatial part, a stochastic part, and a part which quantifies the quadrature error intro-duced by the discrete orthogonal projection. This decomposition paves the way for novelresidual-based adaptive numerical schemes.The article is structured as follows: In Section 2 we describe the problem of interest.In Section 3 the NISP and RKDG method is reviewed and we show how to obtain thereconstruction from our numerical solution. Section 4 presents our main a posteriorierror estimate with decomposition of the residual. Let p Ω , F , P q be a probability space, where Ω is the set of all elementary events ω P Ω , F isa σ -algebra on Ω and P is a probability measure. We consider uncertainties parametrizedby a random variable ξ : Ω Ñ Ξ Ă R with probability density function w ξ : Ξ Ñ R ` .The random variable induces a probability measure ˜ P p B q : “ P p ξ ´ p B qq for all B P B p Ξ q on the measurable space p Ξ , B p Ξ qq , where B p Ξ q is the corresponding Borel σ -algebra.This measure is called the law of ξ and in the following we work on the image probabilityspace p Ξ , B p Ξ q , ˜ P q . For a second measurable space ( E, B p E qq , we consider the weighted L pξ -spaces equipped with the norm } f } L pξ p Ξ; E q : “ $&%´ ş Ξ } f p y q} pE w ξ p y q d y ¯ { p “ E ´ } f } pE ¯ { p , ď p ă 8 ess sup y P Ξ } f p y q} E , p “ 8 . Our problem of interest is the following initial value problem for an one dimensionalsystem of m P N random conservation laws, i.e., B t u p t, x, y q ` B x F p u p t, x, y qq “ , p t, x, y q P p , T q ˆ R ˆ Ξ ,u p , x, y q “ u p x, y q , p x, y q P R ˆ Ξ . (RIVP)Here, u p t, x, y q P U Ă R m is the vector of conserved unknown quantities, F P C p U ; R m q ,is the flux function, u is the uncertain initial condition, U Ă R m is the state space, whichis assumed to be an open set and T P p , describes the final time. We assume that(RIVP) is strictly hyperbolic, i.e. its Jacobian D F p u q has m distinct real eigenvalues.We say that p η, q q P C p U ; R q forms an entropy/entropy-flux pair if η is strictly convexand if η and q satisfy D η D F “ D q . We assume that the random conservation law(RIVP) is equipped with at least one entropy/entropy-flux pair. Following the definitionin [9] for scalar problems, we call u P L ξ p Ξ; L pp , T q ˆ R ; U qq a random entropy solutionof (RIVP), if u p¨ , ¨ , y q is a classical entropy solution, cf. [3, Def. 4.5.1], ˜ P -a.s. y P Ξ .The well-posedness of (RIVP), will not be discussed in this article but can found in [6],2here existence and uniqueness of random entropy solutions for (RIVP) with randomflux functions and random initial data with sufficiently small total variation is proven,based on the results of [1]. For the stochastic discretization of (RIVP) we use the NISP method, [8], which is basedon the (generalized) polynomial chaos expansion which was introduced in [10]. Underthe assumption that u is square-integrable with respect to Ξ , we expand the solution of(RIVP) into a generalized Fourier series using a suitable orthonormal basis.Let t Ψ i p¨qu i P N : Ξ Ñ R be a L ξ p Ξ q -orthonormal basis, i.e. for i, j P N we have A Ψ i , Ψ j E : “ E ´ Ψ i Ψ j ¯ “ ż Ξ Ψ i p y q Ψ j p y q w ξ p y q d y “ δ ij . (3.1)Following [10], the random entropy solution u can be written as u p t, x, y q “ ÿ i “ u i p t, x q Ψ i p y q , (3.2)with (deterministic) Fourier modes u i “ u i p t, x q satisfying u i p t, x q “ E ´ u p t, x, ¨q Ψ i p¨q ¯ @ i P N . (3.3)The NISP method approximates the modes in (3.3) via a discrete orthogonal projection,i.e., numerical quadrature. We denote p R ` q P N quadrature points and weights by t y l u Rl “ , t w l u Rl “ , and approximate u i p t, x q “ ż Ξ u p t, x, y q Ψ i p y q w ξ p y q d y « R ÿ l “ u p t, x, y l q Ψ i p y l q w l “ : ˆ u i for i P N . (3.4)In a second step the NISP method truncates (3.2) after the M -th mode, i.e., u p t, x, y q « M ÿ i “ ˆ u i p t, x q Ψ i p y q . (3.5)For any l “ , . . . , R , the random entropy solution u of (RIVP) evaluated at quadraturepoint t y l u Rl “ , is denoted by u p¨ , ¨ , y l q and it is an entropy solution of the deterministicversion of (RIVP), i.e. of B t u p t, x, y l q ` B x F p u p t, x, y l qq “ , p t, x q P p , T q ˆ R ,u p , x, y l q “ u p x, y l q , x P R . (DIVP q l q l can be discretized by a suitable numericalmethod. We use the RKDG method as described in [2]. We denote the correspondingnumerical solution of (DIVP q l at quadrature point t y l u Rl “ and at points t t n p y l qu N t p y l q n “ , N t p y l q P N , in time by u nh p¨ , y l q P V sp , where V sp : “ t v : R Ñ R m | v | K P P p p K ; R m q , K P T u , is the corresponding DG space of polynomials of degree p P N , associated with a uniformtriangulation T of R . Let us assume that the time partition t t n u N t n “ and the triangulation T used for (DIVP q l are the same for every quadrature point t y l u Rl “ . The numericalapproximation of (RIVP) at time t “ t n can then be written as u nh p x, y q : “ M ÿ i “ ´ R ÿ l “ u nh p x, y l q Ψ i p y l q w l ¯ Ψ i p y q . (3.6)The proof of the a posterior error estimate in Theorem 4.1 uses the relative entropyframework, cf. [3, Section 5.2], which requires one quantity which is at least Lipschitzcontinuous in space and time. To this end we reconstruct the numerical solution sothat we obtain a Lipschitz continuous function. To avoid technical overhead, we do notelaborate upon this process here, but refer to [4, 7], where a detailed description can befound.The reconstruction provides us with a computable space-time reconstruction ˆ u st p y l q P W pp , T q ; V sp ` X C p R qq of the numerical solution t u nh p y l qu N t n “ Ă V sp , for each quadra-ture point t y l u Rl “ . This allows us to define a space-time residual as follows. Definition 3.1 (Space-time residual) . For all l “ , . . . , R , we define R st p y l q P L pp , T qˆ R ; R m q by R st p y l q : “ B t ˆ u st p y l q ` B x F p ˆ u st p y l qq (3.7) to be the space-time residual associated with the quadrature point y l . Next we define the reconstructed mode, the space-time-stochastic reconstruction andthe space-time-stochastic residual. The latter is obtained by plugging the space-time-stochastic reconstruction into the random conservation law (RIVP).
Definition 3.2 (Space-time-stochastic reconstruction and residual) . Let t ˆ u st p y l qu Rl “ : p , T q ˆ R Ñ R m be the sequence of space-time reconstructions at quadrature points t y l u Rl “ . The reconstructed modes of (3.4) are defined as ˆ u sti : “ R ÿ l “ ˆ u st p y l q Ψ i p y l q w l , (3.8) for i “ , . . . , M . The space-time-stochastic reconstruction ˆ u sts : p , T q ˆ R ˆ Ξ Ñ R m is defined as ˆ u sts p t, x, y q : “ M ÿ i “ ˆ u sti p t, x q Ψ i p y q . (3.9)4 inally, we define the space-time-stochastic residual R sts P L ξ p Ξ; L pp , T q ˆ R ; R m qq by R sts : “ B t ˆ u sts ` B x F p ˆ u sts q . (3.10)This residual is crucial in the upcoming error analysis. Before stating the main a posterior error estimate, let us note that derivatives of the fluxfunction and the entropy are bounded on any compcat subset C of the state space. Thesebounds enter the upper bound in Theorem 4.1. Let C Ă U be convex and compact. Dueto F P C p U , R m q and η P C p U , R m q strictly convex there exist constants ă C F ă 8 and ă C η ă C η ă 8 , s.t. | v J HF p u q v | ď C F | v | , C η | v | ď v J Hη p u q v ď C η | v | , @ v P R m , @ u P C . Here HF denotes the Hessian (i.e. the tensor of second order derivatives) of the fluxfunction and Hη the Hessian of the entropy η . We now have all ingredients together tostate the following a posteriori error estimate that can be directly derived from [5]. Theorem 4.1 (A posteriori error bound for the numerical solution) . Let u be the randomentropy solution of (RIVP) . Then, for any n “ , . . . , N t , the difference between u p t n , ¨ , ¨q and the numerical solution u nh from (3.6) satisfies } u p t n , ¨ , ¨q ´ u nh p¨ , ¨q} L ξ p Ξ; L p R qq ď } ˆ u sts p t n , ¨ , ¨q ´ u nh p¨ , ¨q} L ξ p Ξ; L p R qq ` C ´ η ´ E sts p t n q ` C η E sts ¯ ˆ exp ´ C ´ η t n ż ´ C η C F }B x ˆ u sts p t, ¨ , ¨q} L ξ p Ξ; L p R qq ` C η ¯ d t ¯ , with E sts p t n q : “ } R sts p¨ , ¨ , ¨q} L ξ p Ξ; L pp ,t n qˆ R qq , E sts : “ } u p¨ , ¨q ´ ˆ u sts p , ¨ , ¨q} L ξ p Ξ; L p R qq . Proof.
We apply [5, Lemma 5.1] path-wise in Ξ , integrate over Ξ and use Gronwall’s in-equality to bound } u p t n , ¨ , ¨q´ ˆ u sts p t n , ¨ , ¨q} L ξ p Ξ; L p R qq by the second term in the inequality.The final estimate then follows using the triangle inequality.In Theorem 4.1 the error between the numerical solution and the entropy solutionis bounded by the error in the initial condition, the difference between the numericalsolution and its reconstruction and the contribution of the space-time stochastic residual5 sts from (3.10), quantified by E sts . We would like to distinguish between errors that arisefrom discretizing the random space and from discretizing the physical space. Therefore,we show in Lemma 4.1 a splitting of the space-time-stochastic residual R sts into threeparts. Namely a deterministic residual, which corresponds to the spatial error whenapproximating (DIVP q l using the RKDG method, a quadrature residual that reflectsthe quadrature error from the discrete orthogonal projection in (3.4) and a stochasticcut-off error, which occurs when truncating the infinite Fourier series in (3.2). Lemma 4.2 (Orthogonal decomposition of the space-time-stochastic residual) . Thespace-time-stochastic residual R sts from (3.10) admits the following orthogonal decom-position, R sts “ M ÿ j “ ´ R detj ` R sqj ¯ Ψ j ` ÿ j ą M R scj Ψ j , (4.1) where R detj : “ R ÿ l “ R st p y l q Ψ j p y l q w l for j “ , . . . , MR sqj : “ A B x F ´ M ÿ i “ ˆ u st p y i q Ψ i ¯ , Ψ j E ´ R ÿ l “ B x F p ˆ u st p y l qq Ψ j p y l q w l for j “ , . . . , MR scj : “ A B x F ´ M ÿ i “ ˆ u st p y i q Ψ i ¯ , Ψ j E for j ą M are called the j -th mode of the deterministic, stochastic quadrature and stochastic cut-offresidual. Moreover, we have E sts p t q “ } R sts } L ξ p Ξ; L pp ,t qˆ R q “ M ÿ i “ } R deti ` R sqi } L pp ,t qˆ R q ` ÿ i ą M } R sci } L pp ,t qˆ R q ď E det p t q ` E sq p t q ` E sc p t q , (4.2) where, for any t P p , T q , E det p t q : “ M ÿ i “ } R deti } L pp ,t qˆ R q , E sq p t q : “ M ÿ i “ } R sqi } L pp ,t qˆ R q , E sc p t q : “ ÿ i ą M } R sci } L pp ,t qˆ R q . Proof.
We recall that the space-time reconstruction ˆ u st p y l q satisfies R st p y l q “ B t ˆ u st p y l q ` B x F p ˆ u st p y l qq (4.3)6or all l “ , . . . , R . Moreover, the reconstructed mode ˆ u stj was defined as (cf. (3.8)) ˆ u stj “ R ÿ l “ ˆ u st p y l q Ψ j p y l q w l (4.4)for all j “ , . . . , M . Multiplying (4.3) by Ψ j p y l q w l and suming over l “ , . . . , R yields,using (4.4), the following relationship R ÿ l “ R st p y l q Ψ j p y l q w l “ B t ˆ u stj ` R ÿ l “ B x F p ˆ u st p y l qq Ψ j p y l q w l . (4.5)By definition of the space-time-stochastic residual we have R sts “ B t ˆ u sts ` B x F p ˆ u sts q “ B t ´ M ÿ i “ ˆ u sti Ψ i ¯ ` B x F ´ M ÿ i “ ˆ u sti Ψ i ¯ . Let us begin by studying the j -th mode of R sts for j “ , . . . , M . In this case theorthogonality relation (3.1) yields A R sts , Ψ j E “ A B t ˆ u sts ` B x F p ˆ u sts q , Ψ j E “ B t ˆ u stj ` A B x F ´ M ÿ i “ ˆ u sti Ψ i ¯ , Ψ j E . (4.6)Using (4.5) we obtain A R sts , Ψ j E “ R ÿ l “ R st p y l q Ψ j p y l q w l (4.7) ` A B x F ´ M ÿ i “ ˆ u sti Ψ i ¯ , Ψ j E ´ R ÿ l “ B x F p ˆ u st p y l qq Ψ j p y l q w l “ R detj ` R sqj . For j ą M the j -th moment of R sts is A R sts , Ψ j E “ A B x F ´ M ÿ i “ ˆ u sti Ψ i ¯ , Ψ j E “ R scj . (4.8)Formula (4.1) then follows from (4.7) and (4.8). Formula (4.2) is an application of thePythagorean theorem for L ξ p Ξ q .Putting together Theorem 4.1 and Lemma 4.2 we obtain our main result, the followinga posteriori error estimate with separable error bounds. Theorem 4.3 (A posteriori error bound for the numerical solution with error splitting) . Let u be the random entropy solution of (RIVP) . Then, for any n “ , . . . , N t , the ifference between u p t n , ¨ , ¨q and u nh from (3.6) satisfies } u p t n , ¨ , ¨q ´ u nh p¨ , ¨q} L ξ p Ξ; L p R qq ď } ˆ u sts p t n , ¨ , ¨q ´ u nh p¨ , ¨q} L ξ p Ξ; L p R qq ` C ´ η ´ E det p t n q ` E sq p t n q ` E sc p t n q ` C η E sts ¯ ˆ exp ´ C ´ η t n ż ´ C η C F }B x ˆ u sts p t, ¨ , ¨q} L ξ p Ξ; L p R qq ` C η ¯ d t ¯ . We derived a novel residual-based a posteriori error bound for the difference between theentropy solution of (RIVP) and its numerical approximation using the NISP method incombination with a RKDG scheme. Moreover, we proved that the upper bound can bedecomposed into three parts, where E det quantifies the space-time discretization error ofthe RKDG scheme, E sq assesses the quality of the discrete orthogonal projection and E sc quantifies the stochastic error by truncation of the generalized polynomial chaosseries. Based on these results, residual-based adaptive numerical schemes, which balancethe contribution of the three different sources of numerical error, can be constructed.Residual-based space-stochastic adaptive numerical schemes are also considered in [6]. References [1]
A. Bressan and P. LeFloch , Uniqueness of weak solutions to systems of conser-vation laws , Arch. Rational Mech. Anal., 140 (1997), pp. 301–317.[2]
B. Cockburn and C.-W. Shu , Runge-Kutta discontinuous Galerkin methods forconvection-dominated problems , J. Sci. Comput., 16 (2001), pp. 173–261.[3]
C. M. Dafermos , Hyperbolic conservation laws in continuum physics, vol. 325 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math-ematical Sciences], Springer-Verlag, Berlin, fourth ed., 2016.[4]
A. Dedner and J. Giesselmann , A posteriori analysis of fully discrete methodof lines discontinuous Galerkin schemes for systems of conservation laws , SIAM J.Numer. Anal., 54 (2016), pp. 3523–3549.[5]
J. Giesselmann, C. Makridakis, and T. Pryer , A posteriori analysis of dis-continuous Galerkin schemes for systems of hyperbolic conservation laws , SIAM J.Numer. Anal., 53 (2015), pp. 1280–1303.[6]
J. Giesselmann, F. Meyer, and C. Rohde , A posteriori error analysis andnon-intrusive adaptive numerical schemes for systems of random conservation laws. ,arXiv preprint: 1902.05375, (2019). 87] ,
A posteriori error analysis for random scalar conservation laws using thestochastic galerkin method. , IMA J. Numer. Anal., (2019).[8]
O. P. Le Maître, M. T. Reagan, H. N. Najm, R. G. Ghanem, and O. M.Knio , A stochastic projection method for fluid flow. II. Random process , J. Comput.Phys., 181 (2002), pp. 9–44.[9]
N. H. Risebro, C. Schwab, and F. Weber , Multilevel Monte Carlo front-tracking for random scalar conservation laws , BIT, 56 (2016), pp. 263–292.[10]