Most real world models involve a large number of parameters and coefficients which cannot be exactly determind. Furthermore, there is a considerable uncertainty in the source terms, initial or boundary data due to empirical approximations or measuring errors. Therefore, study of PDEs with randomness (stochastic PDEs) certainly leads to greater understanding of the actual physical phenomenon. In this paper, we are interested in a stochastic variant of the compressible barotropic Euler system, a set of balance laws driven by a nonlinear multiplicative noise for mass density and the bulk velocity describing the flow of isentropic gas, where the thermal effects are neglected. The system of equations read
Here denotes the adiabatic exponent, is the squared reciprocal of the Mach number (the ratio between average velocity and speed of sound). The driving process
is a cylindrical Wiener process defined on some filtered probability space, and the noise coefficient is nonlinear and satisfies suitable growth assumptions (see Subsection 2.2 for the complete list of assumptions). Note that is a given Hilbert space valued function signifying the multiplicative nature of the noise. We consider the stochastic compressible Euler equations (1.1)–(1.2) in three spatial dimensions on a periodic domain i.e., on the torus
. The initial conditions are random variables
with sufficient spatial regularity to be specified later.
1.1 Compressible Euler Equations
The deterministic counterpart of the stochastic compressible Euler equations (1.1)–(1.2) have received considerable attention and, in spite of monumental efforts, satisfactory well-posedness results are still lacking. It is well-known that the smooth solutions to deterministic counterpart of (1.1)–(1.2) exists only for a finite lap of time, after which singularities may develop for a generic class of initial data. Therefore, global-in-time (weak) solutions must be sought in the class of discontinuous functions. But, weak solutions may not be uniquely determind by their initial data and admissibility conditions must be imposed to single out the physically correct solution. However, the specification of such an admissibility criteria is still open. Indeed, thanks to recent phenomenal work by De Lellis Szekelyhidi [14, 15], and further investigated by Chiodaroli et. al. , Feireisl , it is well understood that the compressible Euler equations is desparetly ill-posed, due to the lack of compactness of functions satisfying the equations. Even if the initial data is smooth, the global existence and uniqueness of solutions can fail. Moreover, a quest for the existence of global-in-time weak solutions to deterministic counterpart of (1.1)–(1.2) for general initial data remains elusive. Given this status quo, it is natural to seek an alternative solution paradigm for compressible Euler system. To that context, we recall the framework of dissipative Young measure-valued solutions in the context of compressible Navier–Stokes system, being first introduced by Neustupa in , and subsequently revisited by Feireisl et. al. in . In a nutshell, these solutions are characterized by a parametrized Young measure and a concentration Young measure in the total energy balance, and they are defined globally in time.
The study of stochastic compressible Euler equations (1.1)–(1.2) is a relatively new area of focus within the broarder field of stochastic PDEs, and a satisfactory well/ill-posedness result is largely out of reach. However, we want to emphasize that, to design efficient numerical schemes it is of paramount importance to have prior knowledge about the existence of global-in-time solutions for the underlying system of equations. Without such knowledge, there is no way to establish whether or not the solution produced by a numerical scheme is an approximation of the true solution. To that context, let us first mention the work by Berthelin Vovelle , where the authors established the existence of a martingale solution for (1.1)–(1.2) in one spatial dimension. Moreover, a recent work by Breit et. al. in  revealed that ill-posedness issues for compressible Euler system driven by additive noise, in the sense of [14, 22], persist even in the presense of a random forcing. We mention that for compressible Euler equations driven by multiplicative noise, the existence of dissipative measure-valued martingale solutions was very recently established by Hofmanova et. al. in  (see also  for the incompressible case). The authors have shown that the existence can be obtained from a sequence of solutions of stochastic Navier Stokes equations using tools from martingale theory and Young measure theory.
1.2 Numerical Schemes
Parallel to mathematical efforts there has been a huge effort to derive effective numerical schemes for deterministic fluid flow equations, and there is a considerable body of literature dealing with the convergence of numerical schemes for the specific problems in fluid mechanics represented through the barotropic Euler system. In this context, we first mention the work by Karper in  where he has established the convergence of a mixed finite element-discontinuous Galerkin scheme to compressible Euler system under the assumption . Subsequently, a series of works [18, 19, 20] by Feireisl and his collaborators analyzed the convergence issues for several different semi-discrete numerical schemes via the framework of dissipative measure-valued solutions. Note that the concept of measure–valued solutions introduced in Feireisl et. al.  (and also ) requires the solutions generated by approximate sequences satisfying only the general energy bounds. This is very different from many classical approach where the existence of measure-valued solution is conditioned by mostly rather unrealistic assumptions of boundedness of certain physical quantities and the corresponding fluxes. Indeed, assuming only uniform lower bound on the density and uniform upper bound on the energy they showed that the Lax-Friedrichs-type finite volume schemes generate the dissipative measure–valued solutions to the barotropic Euler equations. We also mention that the first numerical evidence that indicated ill-posedness of the Euler system was presented by Elling . Finally, we mention a series of recent works by Fjordholm et. al. [23, 24] in the context of a general system of hyperbolic conservation laws, where they proved the convergence of a semi-discrete entropy stable finite volume scheme to the measure-valued solutions under certain appropriate assumptions.
We remark that, despite the growing interest about the theory of stochastic PDEs and the discretization of stochastic PDEs, the specific question about numerical approximations of stochastic compressible Euler equations is virtually untouched. In fact, the challenges related to numerical aspects of (1.1) are manifold and mostly open, due to the presence of multiplicative noise term in (1.1). Having said this, we mention that there are few results available on stochastic incompressible Euler equations. To that context, concerning the convergence of the numerical methods, we mention the work of Brzeźniak et. al. , where the scheme is based on finite elements combined with implicit Euler method.
1.3 Scope and Outline of the Paper
The above discussions clearly highlight the lack of effective convergent numerical schemes, for compressible fluid flow equations driven by a multiplicative Brownian noise, which are able to take the inherent uncertainties into account, and are equipped with modules that quantify the level of uncertainty. The challenges related to numerical aspects of the underlying problems are mostly open and the research on this frontier is still in its infancy. In fact, the main objective of this article is to lay down the foundation for a comprehensive theory related to numerical methods for (1.1)–(1.2). Although our work bears some similarities with recent wroks of Fjordholm et. al [23, 24] on deteministic system of conservation laws, and works of Feireisl et. al [18, 19, 20] on deterministic Euler systems, the main novelty of this work lies in successfully handling the multiplicative noise term. Our problems need to invoke ideas from numerical methods for SDE and meaningfully fuse them with available approximation methods for deterministic problems. This is easier said than done as any such attempt has to capture the noise-noise interaction as well. In the realm of stochastic conservation laws, noise-noise interaction terms play a fundamental role to establish well-posedness theory, for details see [3, 4, 5, 29, 30, 31, 32].
The main contributions of this paper are listed below:
We develop an appropriate mathematical framework of dissipative measure-valued martingale solutions to the stochastic compressible Euler system, keeping in mind that this framework would allow us to establish weak (measure-valued)–strong uniqueness principle. We remark that our solution framework requires only natural energy bounds associated to approximate solutions.
We show that a Lax-Friedrichs-type numerical scheme for (1.1)–(1.2) generates the dissipative measure-valued martingale solutions to the stochastic compressible Euler equations. With the help of the new framework based on the theory of measure–valued solutions, we adapt the concept of -convergence, first developed in the context of Young measures by Balder  (see also Feireisl et. al. ), to show the pointwise convergence of arithmetic averages (Cesaro means) of numerical solutions to a dissipative measure-valued martingale solution of the limit system (1.1)–(1.2).
When solutions of the limit continuous problem possess maximal regularity, by making use of weak (measure-valued)–strong uniqueness principle, we show unconditional strong -convergence of numerical approximations to the regular solution of the limit systems.
A breif description of the organization of the rest of the paper is as follows: we describe all necessary mathematical/technical framework and state the main results in Section 2. Moreover, we introduce a Lax-Friedrichs-type finite volume numerical scheme for the underlying system (1.1)–(1.2). Section 3 is devoted on deriving stability properties of the scheme, while Section 4 is focused on deriving suitable formulations of the continuity and momentum equations, and exhibit consistency. In Section 5, we present a proof of convergence of numerical solutions to a dissipative measure-valued martingale solutions using stochastic compactness. Section 6 is devoted on deriving the weak (measure-valued) – strong uniqueness principle by making use of a suitable relative energy inequality. Section 7 uses the concept of -convergence to exhibit the pointwise convergence of numerical solutions. Finally, in Section 8, we make use of weak (measure-valued)–strong uniqueness property to show the convergence of numerical approximations to the solutions of stochastic compressible Euler system (1.1)–(1.2).
2 Preliminaries and Main Results
Here we first briefly recall some relevant mathematical tools which are used in the subsequent analysis and then we state main results of this paper. To begin, we fix an arbitrary large time horizon . For the sake of simplicity it will be assumed , since its value is not relevant in the present setting. Throughout this paper, we use the letter to denote various generic constants that may change from line to line along the proofs. Explicit tracking of the constants could be possible but it is highly cumbersome and avoided for the sake of the reader. Let denote the space of bounded Borel measures on whose norm is given by the total variation of measures. It is the dual space to the space of continuous functions vanishing at infinity equipped with the supremum norm. Moreover, let be the space of probability measures on .
2.1 Analytic framework
Let be given, and be a separable Hilbert space. Let denotes a -valued Sobolev space which is characterized by its norm
Then we have following compact embedding result from Flandoli Gatarek [25, Theorem 2.2].
If are two Banach spaces with compact embedding, and real numbers satisfy , then the following embedding
2.1.1 Young measures, concentration defect measures
In this subsection, we first briefly recall the notion of Young measures and related results which have been used frequently in the text. For an excellent overview of applications of the Young measure theory to hyperbolic conservation laws, we refer to Balder . Let us begin by assuming that is a sigma finite measure space. A Young measure from into is a weakly measurable function in the sense that is -measurable for every Borel set in . In what follows, we make use of the following generalization of the classical result on Young measures; for details, see [8, Section 2.8].
Let , and let , , be a sequence of random variables such that
Then on the standard probability space , there exists a new subsequence (not relabeled), and a parametrized family (superscript emphasises the dependence on ) of random probability measures on , regarded as a random variable taking values in , such that has the same law as , i.e. and the following property holds: for any Carathéodory function , such that
In literature, Young measure theory has been successfully exploited to extract limits of bounded continuous functions. However, for our purpose, we need to deal with typical functions for which we only know that
In fact, using a well-known fact that is embedded in the space of bounded Radon measures , we can infer that -a.s.
where , and is called concentration defect measure (or concentration Young measure). We remark that, a simple truncation analysis and Fatou’s lemma reveal that -a.s. and thus -a.s. is finite for a.e. . In what follows, regarding the concentration defect measure, we shall make use of the following crucial lemma. For a proof of the lemma modulo cosmetic changes, we refer to Feireisl et. al [21, Lemma 2.1].
Let , be a sequence generating a Young measure , where is a measurable set in . Let be a continuous function such that
and let be continuous such that
Let us denote -a.s.
Here are weak- limits of , respectively in . Then -almost surely .
2.1.2 Convergence of arithmetic averages
Following Feireisl et. al. , we also show that the arithmetic averages of numerical solutions converge pointwise to a generalized dissipative solution of the compressible Euler system, as introduced in Hofmanova et. al. . To that context, we have the following result.
Let be a finite measure space, and weakly in . Then there exists a subsequence of sequence such that
Since the sequence is uniformly bounded in , thanks to Komlós theorem, there exists a subsequence and such that
Let us define . Since is also converges weakly to , it implies that converges weakly to in . So sequence is uniformly integrable in . As consequence of Vitali’s convergence theorem implies that converges to strongly in . Therefore, uniqueness of weak limit implies that in . This concludes the proof. ∎
2.2 Background on Stochastic framework
Here we briefly recapitulate some basics of stochastic calculus in order to define the cylindrical Wiener process and the stochastic integral appearing in (1.1). To that context, let be a stochastic basis with a complete, right-continuous filtration. The stochastic process is a cylindrical -Wiener process in a separable Hilbert space . It is formally given by the expansion
where is a sequence of mutually independent real-valued Brownian motions relative to and is an orthonormal basis of . To give the precise definition of the diffusion coefficient , consider , , and such that . Denote and let be defined as follows
The coefficients are -functions that satisfy uniformly in
As usual, we understand the stochastic integral as a process in the Hilbert space , . Indeed, it is easy to check that under the above assumptions on and , the mapping belongs to , the space of Hilbert–Schmidt operators from to . Consequently, if111Here denotes the predictable -algebra associated to .
and the mean value is essentially bounded then the stochastic integral
is a well-defined -martingale taking values in . Note that the continuity equation (1.1) implies that the mean value of the density is constant in time (but in general depends on ). Finally, we define the auxiliary space via
endowed with the norm
Note that the embedding is Hilbert–Schmidt. Moreover, trajectories of are -a.s. in .
For the convergence of approximate solutions, it is necessary to secure strong compactness (a.s. convergence) in the -variable. For that purpose, we need a version of Skorokhod representation theorem, so-called Skorokhod-Jakubowski representations theorem. Note that classical Skorokhod theorem only works for Polish spaces, but in our analysis path spaces are so-called quasi-Polish spaces. In this paper, we use the following version of the Skorokhod-Jakubowski theorem, taken from Brzeźniak et.al. .
Let be a complete separable metric space and be a topological space such that there is a sequence of continuous functions that separates points of . Let be a stochastic basis with a complete, right-continuous filtration and be a tight sequence of random variables in , where and is equipped with the topology induced by the canonical projections and . Note that is the -algebra generated by the sequence , .
Assume that there exists a random variable in such that . Then there exists a subsequence and random variables in for on a common probability space with
in almost surely for .
Finally, we mention the “Kolmogorov test” for the existence of continuous modifications of real-valued stochastic processes.
Let be a real-valued stochastic process defined on a probability space . Suppose that there are constants , and such that for all ,
Then there exists a continuous modification of and the paths of are -Hölder continuous for every .
2.3 Stochastic compressible Euler equations
Since we aim at proving pointwise convergence of numerical solutions to the regular solution of the limit system, using the weak (measure-valued)–strong uniqueness principle for dissipative measure-valued solutions, we first recall the notion of local strong pathwise solution for stochastic compressible Euler equations, being first introduced in . Such a solution is strong in both the probabilistic and PDE sense, at least locally in time. To be more precise, system (1.1)–(1.2) will be satisfied pointwise (not only in the sense of distributions) on the given stochastic basis associated to the cylindrical Wiener process .
Definition 2.7 (Local strong pathwise solution).
Let be a stochastic basis with a complete right-continuous filtration. Let be an -cylindrical Wiener process and be a -valued -measurable random variable, for some , and let satisfy (2.1) and (2.2). A triplet is called a local strong pathwise solution to the system (1.1)–(1.2) provided
is an a.s. strictly positive -stopping time;
the density is a -valued -progressively measurable process satisfying
the velocity is a -valued -progressively measurable process satisfying
there holds -a.s.
for all .
Note that classical solutions require spatial derivatives of and to be continuous -a.s. This motivates the following definition.
Definition 2.8 (Maximal strong pathwise solution).
Fix a stochastic basis with a cylindrical Wiener process and an initial condition as in Definition 2.7. A quadruplet
is an a.s. strictly positive -stopping time;
is an increasing sequence of -stopping times such that on the set , a.s. and
each triplet , , is a local strong pathwise solution in the sense of Definition 2.7.
There are quite a few results available in the literature concerning the existence of maximal pathwise solutions for various SPDE or SDE models, see for instance [9, 17]. For compressible Euler equations, a specific work can be found in Breit Mensah in [7, Theorem 2.4].
2.4 Measure-valued solutions
Note that, in general any uniformly bounded sequence in does not immediately imply weak convergence of it due to the presence of oscillations and concentration effects. To overcome such a problem, two kinds of tools are used:
Young measures: these are probability measures on the phase space and accounts for the persistence of oscillations in the solution;
Concentration defect measures: these are measures on physical space-time, accounts for blow up type collapse due to possible concentration points.
2.4.1 Dissipative measure-valued martingale solutions
Keeping in mind the previous discussion, we now introduce the concept of dissipative measure–valued martingale solution to the stochastic compressible Euler system. In what follows, let
be the phase space associated to the Euler system.
Definition 2.10 (Dissipative measure-valued martingale solution).
is a random variable taking values in the space of Young measures on . In other words, -a.s. is a parametrized family of probability measures on ,
is a stochastic basis with a complete right-continuous filtration,
is a -cylindrical Wiener process,
the average density satisfies for any -a.s., the function is progressively measurable and
for all ,
the average momentum satisfies for any -a.s., the function is progressively measurable and
for all ,
the integral identity
holds -a.s., for all , and for all ,
the integral identity
holds -a.s., for all , and for all , where ,
-a.s., is a tensor–valued measure,
there exists a real-valued martingale , such that the following energy inequality
holds -a.s., for all in with
Here , , -a.s., , , -a.s., with initial energy.
there exists a constant such that
holds -a.s., for every .
We remark that, in light of a standard Lebesgue point argument applied to (2.7), energy inequality holds for a.e. in :
However, to establish weak (measure-valued)–strong uniqueness principle, we require energy inequality to hold for all . This can be achieved following the argument depicted in Section 5.
2.5 Numerical scheme
It is well known that standard finite difference, finite volume and finite element methods have been very successful in computing solutions to system of hyperbolic conservation laws, including deterministic compressible fluid flow equations. Here we consider a semi-discrete finite volume scheme for the stochastic compressible Euler equations (1.1)–(1.2). In what follows, drawing preliminary motivation from the analysis depicted in [18, 19, 20], we describe the finite volume numerical scheme which is later shown to converge in appropriate sense. More precisely, we show that the sequence of numerical solutions generate the Young measure that represents the dissipative measure-valued martingale solution.
2.5.1 Spatial discretization
We begin by introducing some notation needed to define the semi-discrete finite volume scheme. Throughout this paper, we reserve the parameter to denote small positive numbers that represent the spatial discretizations parameter of the numerical scheme. Note that, since we are working in a periodic domain in , the relevant domain for the space discretization is . To this end, we introduce the space discretization by finite volumes (control volumes). For that we need to recall the definition of so called admissible meshes for finite volume scheme.
Definition 2.13 (Admissible mesh).
An admissible mesh of is a family of disjoint regular quadrilateral connected subset of satisfying the following:
is the union of the closure of the elements (called control volume K) of , i.e., .
The common interface of any two elements of
is included in a hyperplane of.
There exists nonnegative constant such that
where , denotes the -dimensional Lebesgue measure of , and represents the -dimensional Lebesgue measure of .
In the sequel, we denote the followings:
: the set of interfaces of the control volume .
: the set of control volumes neighbors of the control volume .
: the common interface between and , for any .
: the set of all the interfaces of the mesh .
: the unit normal vector to interface, oriented from to , for any .
: the unit basis vector in the -th space direction, Note that in our case the mesh is a regular quadrilateral grid, and thus is parallel to , for some
Let denote the space of piecewise constant functions defined on admissible mesh For we set Then it holds that
The value of on the face shall be denoted by and analogously for faces of cell in direction. We also introduce a standard projection operator
For we define the following discrete operators
The discrete Laplace and divergence operators are defined as follows
Furthermore, on the face we define the jump and mean value operators
respectively. Here denote the unit outer normal to and respectively. Finally, we introduce the mean value of in cell in the direction of by
2.5.2 Entropy stable flux and the scheme
Note that constructing and analyzing numerical schemes for the deterministic counterpart of the underlying system of equations (1.1)–(1.2) has a long tradition. Usually the schemes are developed to satisfy certain additional properties like entropy condition and kinetic energy stability which can be important for turbulent flows. To that context, Tadmor  proposed the idea of entropy conservative numerical fluxes which can then be combined with some dissipation terms using entropy variables to obtain a scheme that respects the entropy condition, i.e., the scheme must produce entropy in accordance with the second law of thermodynamics. Such a flux is called entropy stable flux.
where we introduced the variables , , and .
Note that (2.10) is a stochastic differential equation in . Let us now specify the numerical flux associated to the flux function . Indeed, we want to satisfy the following properties:
(Consistency) The function satisfies , for all .
(Lipschitz continuity) There exist two constants such that for any , it holds that