Simplified Hyperbolic Moment Equations

Transcription

1 Simplified Hyperbolic Moment Equations Julian Koellermeier and Manuel Torrilhon Abstract Hyperbolicity is a necessary property of model equations for the solution of the BGK equation to achieve stable and physical solutions. However, the standard approach for velocity space discretization developed by Grad only yields locally hyperbolic equations. The method has recently been improved and several new globally hyperbolic model systems have been derived such as the Hyperbolic Moment Equations () and the Quadrature-Based Moment Equations (). We will describe the derivation and properties of a new model system called Simplified Hyperbolic Moment Equations (S) which inherits hyperbolicity from the other models but is simpler to implement and to solve. First simulation results show a good accuracy of the new S model in comparison to the existing models. Introduction There are several possible solution methods for the BGK Equation with varying success. Among those are direct solvers like discrete velocity methods [2], particle methods like DSMC [2] and moment methods as for example []. A relatively old moment approach was developed by Grad in [6] but due to problems regarding the loss of hyperbolicity of the equations, there has not been much research on this approach for a long time. Hyperbolicity is an important property because otherwise there will be imaginary eigenvalues creating instabilities and non-physical solutions. Grad s equations have been shown to be only hyperbolic in a small region around equilibrium so that the solution can break down for strong non-equilibrium, see []. Julian Koellermeier MathCCES, RWTH Aachen University, Schinkelstrasse 2, 262 Aachen koellermeier@mathcces.rwth-aachen.de Manuel Torrilhon MathCCES, RWTH Aachen University, Schinkelstrasse 2, 262 Aachen mt@mathcces.rwth-aachen.de

2 2 Julian Koellermeier and Manuel Torrilhon There has been a lot of work regarding different hyperbolic approaches like the maximum entropy method by Levermore [] and the method gives very accurate results, but is also extremely complex because it requires the solution of a non-linear optimization problem in every step. Other methods like the Pearson-IV model developed by Torrilhon [4] seem to be difficult to generalize to the multi-dimensional case. Recently, several new hyperbolic moment models have been developed, that are based on Grad s approach but modify the system of equations in different ways to achieve global hyperbolicity of the equations. Two examples are the Hyperbolic Moment Equations () by Cai [] and the Quadrature-Based Moment Equations () by Koellermeier [8]. As both new models only approximate the original system, it is still necessary to investigate these models with respect to accuracy in various situations. Furthermore, the development of other models is possible, for example using the Operator Projection approach as explained in []. In this paper we present a new hyperbolic moment model called Simplified Hyperbolic Moment Equations (S) that we derive using a special approximation during Grad s method. The new model equations can be explicitly derived and we also show that the model is globally hyperbolic as a special linearization of Grad s equations around equilibrium. The paper is organized as follows: We first recall the BGK Equation, the derivation of the moment method and some existing hyperbolic models in Section 2 before we derive the new S model in the main Section. We show some shock tube results in Section 4 and the paper ends with a conclusion. 2 Moment Method for the BGK Equation In one spatial dimension the BGK Equation describing the change of the particle distribution function f (t,x,c) reads as follows t f (t,x,c) + c f (t,x,c) = S( f ), () x where we will consider the BGK collision operator [] with relaxation time τ on the right-hand side and the equilibrium Maxwellian is given by ( f M (t,x,c) = ρ(t,x) exp 2πθ(t,x) S( f ) = τ ( f f M) (2) ) (c u(t,x))2. () 2θ(t,x) The macroscopic quantities density ρ, velocity u and temperature θ can be computed via integration of the distribution function over velocity space

3 Simplified Hyperbolic Moment Equations ρ(t,x) = f (t,x,c) dc, (4) R ρ(t,x)u(t,x) = c f (t,x,c) dc, () R ρ(t,x)θ(t,x) = c u 2 f (t,x,c) dc. (6) R The solution of () is particularly difficult as it requires an additional discretization of the velocity space. As an efficient way to perform this we will use the moment method. This method requires different steps that have been outlined previously e.g. in [] and we will recall these steps here to derive a new simplified model later.. Expansion of the distribution function The distribution function f (t,x,c) is first expanded in velocity space in a series of basis functions φ [u(t,x),θ(t,x)] α as follows f (t,x,c) = f α (t,x)φ α [u(t,x),θ(t,x)] α N ( ) c u θ () = f α (t,x)φ α [u,θ] (ξ ) (8) α N with new velocity variable ξ = c u θ, (9) see remark below. Furthermore, the superscripts mean that the basis function will depend on the macroscopic quantities u(t,x),θ(t,x). From now on we will omit the arguments t,x in u and θ to shorten notation. Additionally, we use Einstein s summation notation wherever possible to abbreviate the sum in (8). 2. Definition and properties of the basis functions The basis functions are defined as weighted Hermite polynomials according to φ [u,θ] α (ξ ) = 2π θ α+ 2 H α (ξ )exp where H α is the Hermite polynomial of degree α. ( ξ 2 2 ), () Remark. The argument ξ = c u θ can be seen as a transformed velocity variable in which the expansion is performed. The microscopic velocity c is shifted by its mean u and scaled by its variance θ such that the new velocity variable ξ is normalized with variance and mean. This requires less basis functions for the velocity discretization later but leads to a more complicated PDE to discretize. In order to derive the moment equations, we will need to compute derivatives of the basis functions which in turn need the computation of the Hermite derivatives. We note that the Hermite polynomials fulfill the following formulas:

4 4 Julian Koellermeier and Manuel Torrilhon Recursion relation: H α+ (x) = xh α (x) αh α (x) () Derivative: H α(x) = αh α (x), (2) Weighted derivative:[h α (x)exp( x 2 /2)] = αh α+ (x)exp( x 2 /2) () With some basic calculations it is now possible to verify that the basis functions φ α [u,θ] (ξ ) satisfy Recursion relation: θξ φ [u,θ] α (ξ ) = θφ [u,θ] α+ [u,θ] (ξ ) + αφ (ξ ), (4) Derivative: ξ φ α [u,θ] (ξ ) = θφ [u,θ] α+ (ξ ), () θ derivative: θ φ [u,θ] α (ξ ) = α + 2θ α φ [u,θ] α (ξ ). (6). Compatibility constraints According to the definition of the macroscopic quantities, the following conditions can be derived by inserting the ansatz () into (4)-(6) f = ρ, f = f 2 =. () This constrains the unknown coefficients f α to the subspace where the macroscopic quantities are recovered. 4. Derivation of the moment equations The derivation of the moment equations in general form now only needs the computation of the terms in the BGK equation (). The terms t f and x f are computed in the following way for s = x,t s f (t,x,c) = ( ) f α (t,x)φ α [u,θ] (ξ ) s with φ [u,θ] α (ξ ) s = φ[u,θ] = f α(t,x) s where the remaining unknown term is given by Using the previously computed formulas we get φ [u,θ] α (ξ ) + f α (t,x) φ[u,θ] α (ξ ) s (8) α (ξ ) θ θ s + φ[u,θ] α (ξ ) ξ ξ s, (9) ξ s = u θ x ξ θ 2θ s. (2) s f (t,x,c) = f α s φ α [u,θ] (ξ ) + u s f αφ [u,θ] α+ (ξ ) + θ 2 s f αφ [u,θ] α+2 (ξ ) (2) or equivalently using an index shift in the Einstein notation of the infinite sum f α φ [u,θ] α

5 Simplified Hyperbolic Moment Equations ( s f (t,x,c) = fα s + u s f α + ) θ 2 s f α 2 φ α [u,θ] (ξ ). (22) The convective term c x f (t,x,c) can now be computed using (9), (4) and (22) c (u x f (t,x,c) = + ) θξ x f (t,x,c) ( = φ α [u,θ] (ξ ) θ f α + u f α x x + (α + ) f α+ x + u x (θ f α 2 + u f α + (α + ) f α ) (2) + ) θ 2 x (θ f α + u f α 2 + (α + ) f α ). Right-hand side collision term The right-hand side collision term is simply computed by inserting expansion () into (2). Due to the definition of the equilibrium Maxwellian (), we get S( f ) = δ α τ f α φ [u,θ] α, with δ α = {, α =, otherwise. (24) 6. Matrix form of the moment system Cutting off the expansion () at M N, we get M + unknowns that we write as w M = (ρ,u,θ, f,... f M ). The moment system for the unknown vector w M can be directly obtained by matching coefficients of the basis functions in (22), (2) and (24). We can write the system in the following form w M t where the right-hand side reads + A Grad w M x = S M, (2) S M = τ Pw M (26) for diagonal matrix P = diag(,,,,...,). The explicit expressions for (2) can be found in []. For the famous -moment case the system matrix reads u ρ θ ρ u A Grad = 2θ u ρ 6. (2) ρθ 4 f 2 u 4 f θ f ρ f 4 2 θ u

6 6 Julian Koellermeier and Manuel Torrilhon Unfortunately, it has been shown that the system (2) looses hyperbolicity for already moderate non-equilibrium values, see e.g. []. This can lead to non-physical values and a breakdown of the solution as exemplified in []. It is thus of major importance to derive models with unbounded hyperbolicity regions. 2. Existing Hyperbolic Moment Models Two existing moment models that are globally hyperbolic have been derived in [] and []. The Hyperbolic Moment Equations () and the Quadrature-Based Moment Equations () will be used as comparison for later computations in Section 4. For the moment test case the system matrices are given by u ρ v ρ θ ρ u θ ρ v 6 A = 2θ u ρ 6, A ρθ = 2θ v ρ. ρθ 4 f 2 u 4 4 f f 2 f 4 θ v 4 θ ρ f θ u f θ ρ f 4 f θ+ f 4 ρθ v (28) Both models are globally hyperbolic due to the marked changes in the system matrix with respect to Grad s model and they have recently been summarized in a framework for the derivation of hyperbolic moment models in []. A New Simplified Hyperbolic Moment Model S In order to derive efficient but simple moment models to accurately capture flow phenomena beyond the standard fluid dynamics equations we aim to derive new models that overcome difficulties of the standard Grad model. As as result of the derivation of Grad s equations the problematic loss of hyperbolicity is caused by the effects of the higher order coefficients in the equations. These coefficients enter the equation system because the basis functions depend on the shifted velocity variable ξ and thus on u and θ. This dependence is reasonable as it yields an efficient approximation but it effectively spoils the hyperbolicity of the moment equations. In order to reduce the non-linearity introduced by the complicated choice of the basis functions, we will reduce the model complexity using the following approximation to Equation (9), where s = x,t φ [u,θ] α (ξ ) s α (ξ ) θ θ s + φ[u,θ] α (ξ ) ξ ξ s = φ[u,θ], (29) meaning that the derivative of the basis function with respect to time and space is set to zero, which effectively means that the basis functions are treated as if they

7 Simplified Hyperbolic Moment Equations only depended on a fixed velocity space. This leads to large simplifications as it will cancel most terms containing derivatives with respect to the coefficients f α. However, we must make sure not to change the conservation laws of mass, momentum and energy. This is ensured by applying the approximation only to the last M 2 equations, while keeping the first three equations as before. From a physical point of view, the model can be seen as a linearization of the full model (2), which is extremely non-linear due to the expansion in the transformed variable (9). By neglecting the respective terms in (29), this non-linearity is reduced, see also Sections. and.2 for further interpretations of the simplification in (29). The new model is called Simplified Hyperbolic Moment Equations (S), due to the simplification made in (29). The model results in the following system w M t and in the five moment case we obtain the system matrix u ρ θ ρ u A S = 2θ u 6 ρ ρθ 2 u 4 θ u + A S w M x = S M, () and for M > 4 the matrix is a consistent extension of the tridiagonal matrix in (). Note that the system matrix does not depend on the higher order coefficients any more and is tridiagonal which leads to a reduction of complexity when it comes to implementing numerical schemes. However, it is important to analyze the effect of this simplification with respect to model accuracy and hyperbolicity as well. (). Discussion of the S Model Comparing the matrix () with the original system matrix (2) we see that the S system is exactly the Grad system evaluated at equilibrium, i.e. all higher order coefficients in the matrix are set to zero. S can therefore be seen as the linearization of the original Grad system around equilibrium. The characteristic polynomial of the S system matrix () can thus be computed analogously to Grad s system at equilibrium from which we can directly conclude that the S model is globally hyperbolic, as all eigenvalues of () are real. The linearization of Grad s system around equilibrium might sound too simple but it is actually very similar to the approach by Cai et al. in []. In [9] the system was written in convective variables and it was shown that the system matrix in this variables is nothing else than the convective Grad matrix at equilibrium. In the same way the model can be written as a linear deviation from the

8 8 Julian Koellermeier and Manuel Torrilhon convective Grad system. In that sense the S model is just another reasonable approximation of Grad s system in the original set of variables..2 Relation to Discrete Velocity Model A different approach to achieve a very simple model for the solution of the BGK equation () is the Discrete Velocity Method () [2]. It uses point evaluations of the BGK equation at fixed velocity points c i R,i =,...,M to discretize the equation in velocity space as follows t f i + c i x f i = τ ( f i f M (c i )) (2) This is computationally highly efficient as there is no velocity transformation that results in more complicated equations. The system matrix is just a diagonal matrix A = diag(c,...,c M ) with the discrete velocities on the diagonal. However, far more velocity points are needed to accurately capture the flow, especially for varying mean velocities in the flow field or over time. As one example, more than 4 velocities were used to compute the reference solution Section 4 in comparison with and that used only between five and ten variables. Without the approximation in (29) the basis functions enable a very efficient approach for the discretization in velocity space as the basis effect of the transformed velocity space yields physical adaptivity and also the effect of the derivative is exactly taken into account by (9). The new S model on the other hand simplifies the method in that the derivative of the basis function is neglected, see (29). This reduces the adaptivity of the method. However, the transformation of the velocity variable (9) is still used in all other steps of the derivation such as in the expansion () and during the computation of the term c x f (t,x,c), where the velocity c is substituted by the transformation rule as c = u + θξ. S is thus still adaptive but simply neglects some of the non-linear effects of the adaptivity. We can say that the new S method is in the middle between the standard Grad approach and the method. 4 Simulation Results We test the accuracy of the new S model using a shock tube test case as also done for the and models in previous papers, see e.g. []. The model equation reads t w + A x w = Pw, () τ

9 Simplified Hyperbolic Moment Equations 9 where the system matrix varies depending on the model used. Using M 4 we solve for variables w = (ρ,u,θ, f, f 4,..., f M ). The collisions are modelled using a BGK operator with non-linear relaxation time τ = Kn ρ that leads to the following form of the matrix P P = diag(,,,,,...) R (M+) (M+). (4) We will consider two Knudsen numbers Kn =. and Kn 2 =.. The initial condition is given by { w L, if x < w(,x) = w R, if x > and according to the tests by Cai et al. [] the left and right states are chosen as () w L = (,,,,,...,) T, w R = (,,,,,...,) T, (6) corresponding to a jump in density at the initial discontinuity at x =. For the spatial discretization, we used 4 cells in the domain [ 2,2] and the results show the solution at t END =. using a constant t =.. The numerical scheme to solve the non-conservative PDE system is the PRICE scheme of Canestrelli [4] that was also used in []. The numerical results for the S method in comparison to the other moment models, and a reference solution are shown in Figures and 2. For Kn =. the results are almost identical to the and results. There are only small differences with respect to the other methods. The approximation quality is good even for larger M. In the case of Kn =. in Figure we see that the S model is between and for M = 4. However, the differences are larger for a larger number of moments M. This is as expected and due to the fact that more and more coefficients are neglected in the S approximation when increasing M. Still, the model yields reasonably good results for small M, especially regarding the simplicity of the model. The S solution is not far from the reference solution. Conclusion In this paper we derived a new hyperbolic moment model for the BGK Equation based on the approximation of several non-linear terms during the derivation. The new model equations called Simplified Hyperbolic Moment Equations (S) have been shown to be globally hyperbolic. We compared the model with the discrete velocity model and motivated the use of the new equations by linearization of Grad s equations keeping as much of the adaptivity as possible. The results have shown that the model accuracy is good despite the reduced complexity and the simplified derivation. In order to characterize the new model in more detail additional

10 Julian Koellermeier and Manuel Torrilhon S S (a) M = 4... (b) M = S S (c) M = 6... (d) M = S S (e) M = 8... (f) M = 9 Fig. : Moment model comparison for S,, and reference solution, Kn =.. The left y-axis is for ρ and p, the right y-axis is for u.

11 Simplified Hyperbolic Moment Equations S S (a) M = 4... (b) M = S S (c) M = 6... (d) M = S S (e) M = 8... (f) M = 9 Fig. 2: Moment model comparison for S,, and reference solution, Kn =.. The left y-axis is for ρ and p, the right y-axis is for u.

12 2 Julian Koellermeier and Manuel Torrilhon investigations are necessary for example regarding the stability, convergence and more complex multi-dimensional test cases. References. P. L. Bhatnagar, E. P. Gross, and M. Krook. A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94(): 2, G. A. Bird. Direct simulation and the Boltzmann equation. Physical Fluids, : , 9.. Z. Cai, Y. Fan, and R. Li. Globally hyperbolic regularization of Grad s moment system in one dimensional space. Commun. Math. Sci., (2):4, A. Canestrelli. Numerical Modelling of Alluvial Rivers by Shock Capturing Methods. PhD thesis, Universita Degli Studi di Padova, 28.. Y. Fan, J. Koellermeier, J. Li, R. Li, and M. Torrilhon. Model reduction of kinetic equations by operator projection. J. Stat. Phys., 62(2):4 486, H. Grad. On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2(4): 4, J. Koellermeier. Hyperbolic approximation of kinetic equations using quadrature-based projection methods. Master s thesis, RWTH Aachen University, J. Koellermeier, R. Schaerer, and M. Torrilhon. A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods. Kinet. Relat. Mod., (): 49, J. Koellermeier and M. Torrilhon. On new hyperbolic moment models for the boltzmann equation. In Conference Proceedings of the YIC GACM 2, Publication Server of RWTH Aachen University, 2.. J. Koellermeier and M. Torrilhon. Numerical study of partially conservative moment equations in kinetic theory. Commun. Comput. Phys., 2(4):98, 2.. C. D. Levermore. Moment closure hierarchies for kinetic theories. Journal of Statistical Physics, 8:2 6, L. Mieussens. Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics. Math. Models Methods Appl. Sci., (8):2 49, 2.. H. Struchtrup and M. Torrilhon. Regularization of Grad s moment equations: Derivation and linear analysis. Phys. Fluids, (9): , M. Torrilhon. Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions. Commun. Comput. Phys., (4):69 6, 2.