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Non-Linear Petrov-Galerkin Methods for Reduced OrderHyperbolic Equations and Discontinuous Finite Element
Methods
F.Fanga, C.C.Paina, I.M. Navonb,∗, A.H. Elsheikha, J. Dua,c, D.Xiaoa
aApplied Modelling and Computation Group,Department of Earth Science and Engineering,
Imperial College London,Prince Consort Road, London, SW7 2BP, UK.URL: http://amcg.ese.imperial.ac.uk
bDepartment of Scientific Computing,Florida State University,
Tallahassee, FL, 32306-4120, USAcThe Institute of Atmospheric Physics, the Chinese Academy of Sciences
Beijing, China
Abstract
A new Petrov-Galerkin approach for dealing with sharp or abrupt field changes in dis-continuous Galerkin (DG) reduced order modelling (ROM) is outlined in this paper.This method presents a natural and easy way to introduce a diffusion term into ROMwithout tuning/optimising and provides appropriate modeling and stabilisation for thenumerical solution of high order nonlinear PDEs. The approach is based on the useof the cosine rule between the advection direction in Cartesian space-time and thedirection of the gradient of the solution. The stabilization of the proper orthogonaldecomposition (POD) model using the new Petrov-Galerkin approach is demonstratedin 1D and 2D advection and 1D shock wave cases. Error estimation is carried out forevaluating the accuracy of the Petrov-Galerkin POD model. Numerical results showthe new nonlinear Petrov-Galerkin method is a promising approach for stablisation ofreduced order modelling.
Keywords: Finite Element, Petrov-Galerkin, Proper orthogonal decomposition,Reduced order modelling, Shock wave.
1. Introduction
Reduced Order Model (ROM) technology is a rapidly growing discipline, with sig-nificant potential advantages in: interactive use, emergency response, ensemble cal-culations and data assimilation [1, 2, 3, 4]. ROM is expected to play a major role in
∗Corresponding authorEmail address: [email protected] (I.M. Navon)
Preprint submitted to Elsevier March 13, 2012
facilitating real-time turn-around with computational results and data assimilation. Allmodel reduction methods can be viewed as approximation methods by projection (forcomprehensive description see [5, 6]). Most of those methods (e.g., balanced trunca-tion) are designed for stable, linear and moderate-order systems (state orders are lessthan 104) [7], so are not practical for many fluids systems although they can provide ac-curate low-order representations of state-space systems. Among existing approaches,the proper orthogonal decomposition (POD) method provides an efficient means of de-riving the reduced basis for nonlinear partial differential equations (PDEs). Using thePOD technique, it is possible to extract a set of modes characteristic of the databasewhich constitutes the optimal basis for the energetic description of the flow. A Galerkinprojection of the original equations onto a finite number of POD bases yields a set ofordinary differential equations in time. However, due to the energetic optimality of thePOD bases, only few modes are sufficient to give a good representation of the kineticenergy of the flow. The leading POD modes are not able to dissipate enough energysince the main amount of viscous dissipation takes place in the small eddies (unsolvedmodes) [9]. Galerkin POD methods may thus suffer from a lack of numerical stabilityespecially for high order nonlinear PDEs. There are various ways to recover the effectof the truncated bases (usually the small scales, i.e. unsolved modes) and improve thenumerical stability by:
1. incorporating gradients as well as function values in the definition of POD [10,11];
2. adding calibrated/diffusion terms (e.g. eddy-viscosities,subgrid-scale model) intothe POD reduced order equations [12];
3. the approach of Noack et al. 2009, that uses a finite-time thermodynamics for-malism;
4. the Discrete Empirical Interpolation Method (DEIM) [13] that provides a di-mension reduction of the nonlinear term that has complexity proportional to thenumber of reduced variables by replacing the non-linear terms with a coefficient-function approximation consisting of a linear combination of pre-computed basisfunctions and parameter-dependent coefficients;
5. residual based stablisation approach [9] where the unsolved terms are repre-sented by a number of residual modes which are calculated by the residuals ofROMs [13].
However the main drawback of the above stabilization methods is that there are alwayssome parameters to tune/optimise for a best match the full solution. More recently, thePetrov-Galerkin method has been introduced to POD and applied to the 1D non-linearstatic problem and ODE [14]. This method presents a natural and easy way to intro-duce a diffusion term into ROM without tuning/optimising and provides appropriatemodeling and stabilizations for the numerical solution of high order nonlinear PDEs.In this paper, a new Petrov-Galerkin POD method is presented for non-linearity discon-tinuous Galerkin modelling in order to control numerical oscillations. The approach isbased on the use of the cosine rule between the advection direction in Cartesian space-time and the direction of the gradient of the solution.The remainder of this paper is organized as follows. Section 2 provides the derivationof the new Petrov-Galerkin approach for one scalar time dependent transport equation.
2
This is then followed by the extension to coupled time dependent equations and anexample is provided by two-time level discrete equations in section 3. Section 4 ad-dresses the issue of how how stable reduced order modelling is performed using thenew Petrov-Galerkin approach. In section 5, a Bassi Rebay representation of discon-tinuous Galerkin methods for the diffusion term is described. The method is appliedto 1D and 2D advection and shock cases in section 6 followed by discussion of thenumerical results. Finally, conclusions are drawn in section 7.
2. Scalar Equation
2.1. Non-Linear Petrov-Galerkin scalar equationThe one scalar time dependent transport equation assumes the form:
axt · ∇xtψ + σψ = s, (1)
where ψ represents field states (e.g. temperature, pollutants); s is the source term; σis the reaction term; axt = (at a)T and a = (ax ay az)T (here, the velocity vector,at = ∂a
∂t , ax = ∂a∂x , ay = ∂a
∂y , az = ∂a∂z ). For simplicity, this equation (1) in 1D with time
dependence becomes:
at∂ψ
∂t+ ax
∂ψ
∂x+ σψ = s. (2)
Using the cosine rule between the two vectors axt and ∇xtψ:
cos(θa) =axt · ∇xtψ
|axt | |∇xtψ|, (3)
where θa is the angle between the two vectors axt and ∇xtψ. The projection of axt onto∇xtψ can be written: a∗xt = |axt |nacos(θa) (here na =
∇xtψ|∇xtψ|
). Taking into account (3), itis then written:
a∗xt =(axt · ∇xtψ)∇xtψ
||∇xtψ||22
, (4)
where || · || represents the L2 norm of a vector. Thus
a∗xt · ∇xtψ = axt · ∇xtψ. (5)
Using the Petrov-Galerkin approach, a modified form of the differential equation (2)can be written:
(1 − ∇xt · a∗xt p∗xt)(axt · ∇xtψ + σψ − s) = 0, (6)
where the scalar p∗xt is a function of a∗xt and the size and shape of the elements (see (13),(14) and (9)). Multiplying equation (6) by a space-time basis function Nxti at node i andintegrating it by parts over a single element VE with boundary ΓE (here DG methodsare employed), the discrete form of the scalar equation is written:∫
VE
NxtirdV −∫
ΓE
Nxti(nxt · axt)−(ψ − ψbc)dΓ
3
+
∫VE
(∇xtNxti) · a∗xt p∗xtrdV −
∫ΓE
Nxtinxt · a∗xt p∗xtrdΓ = 0 (7)
with a finite element expansion ψ =∑N
j=1 Nxt jψ j and r = axt · ∇xtψ + σψ − s and nxt isthe normal to the element in space-time and (nxt · axt)− = min0,nxt · axt enables theincoming boundary information to be defined. Applying a zero boundary condition forthe residual r = 0 on ΓE , results in:
∫VE
NxtirdV −∫
ΓE
(nxt · axt)−Nxti(ψ − ψbc)dΓ +
∫VE
(∇xtNxti) · a∗xt p∗xtrdV = 0. (8)
Using the finite element space-time Jacobian matrix Jxt, the scalar p∗xt in (6) can becalculated [18]:
p∗xt =14
(||J−1xt a∗xt ||2)−1. (9)
The finite element space-time Jacobian matrix for 3D time dependent problems is:
Jxt =
∂t∂t′
∂x∂t′
∂y∂t′
∂z∂t′
∂t∂x′
∂x∂x′
∂y∂x′
∂z∂x′
∂t∂y′
∂x∂y′
∂y∂y′
∂z∂y′
∂t∂z′
∂x∂z′
∂y∂z′
∂z∂z′
, (10)
where the variables with ′ are the local variables and where I is the M ×M identitymatrix in whichM is the number of solution variables at each DG node. For uniformspace-time resolution with a time step size of ∆t and an element size of ∆x (in thex-direction), ∆y (in the y-direction), ∆z (in the z-direction), then:
Jxt =
12 ∆t 0 0 00 1
2 ∆x 0 00 0 1
2 ∆y 00 0 0 1
2 ∆z
. (11)
The value of p∗xt can be adjusted to ensure that the resulting value of p∗xt is not so largethat there is more transport backwards than forwards in the resulting discrete system ofequations using:
p∗xt = min1
σ + ε,
14
(||J−1xt a∗xt ||2)−1. (12)
in which σ + ε > 0 and ε is a tolerance used to avoid a ”division by zero” error, hereε = 1 × 10−10. Continuous Petrov-Galerkin formulations use a factor of 1
2 instead of14 in the above. This correctly centers the equation residual at the centre of mass ofthe basis function, for continuous finite element representations. In the present work,where discontinuous finite elements are used to formulate the space-time discretisation,the centre of mass of the basis function is centered at a distance of ∆x
4 from the upwindboundary of the element. In the traditional Petrov Galerkin method a∗xt = axt in theabove and pxt = 1
4 (|Axt · ∇xtNxti|)−1 replaces p∗xt.
4
An approximate expression to p∗xt is obtained from the Riemann finite element method(for details, see [17]):
p∗xt =14
(|a∗xt · ∇xtNxti|)−1. (13)
This will pick up the shape function which is aligned with the direction of a∗xt at leastfor elements with equal sized edges. Alternatively one can produce an p∗xt that is inde-pendent of i with:
p∗xt = mink14
(|a∗xt · ∇xtNxtk |)−1. (14)
Note that this expression uses the length scale of the element in the direction of a∗xt.In the above we can work with the stabilization in the diffusion form with:
∫VE
NxtirdV −∫
ΓE
Nxti(nxt · axt)−(ψ − ψbc)dΓ +
∫VE
(∇xtNxti)Tν∇xtψdV = 0, (15)
where ψbc represents the value of ψ at the boundary, and the scalar diffusion coefficientassumes the form:
ν =(axt · ∇xtψ)p∗xtr||∇xtψ||2
. (16)
The diffusion coefficient ν can be modified to ensure positive diffusion with:
ν =max0, (axt · ∇xtψ)p∗xtr
||∇xtψ||2, (17)
or working only with the residual by replacing axt · ∇xtψ with the residual r whichresults in:
ν =rp∗xtr||∇xtψ||2
. (18)
The diffusion coefficient ν is always non-negative because p∗xt is non-negative. Equa-tion (18) for the diffusivity can be derived by re-defining a∗xt in (4) to be:
a∗xt =r∇xtψ
||∇xtψ||2. (19)
Note that then a∗xt · ∇xtψ =r∇xtψ||∇xtψ||2
· ∇xtψ = r.
2.2. Simplified scalar equationsUsing the two level time-stepping θ-method the residual is:
r = atψn+1 − ψn
∆t+ a · ∇ψn+θ + σψn+θ − sn+θ, (20)
where θ = 0.5, ψn+θ = θψn+1 + (1 − θ)ψn (θ ∈ [0, 1]) and
∇xtψ = (ψn+1 − ψn
∆t, (∇ψn+θ)T )T , (21)
5
which enables the formalism of space-time discretisation to be applied, for example:
a∗xt = (a∗t , a∗T )T =(axt · ∇xtψ)∇xtψ
||∇xtψ||22
, (22)
and
p∗xt = min1
σ + ε,
14
(||J−1a∗||2)−1, (23)
in which J is the block part of the matrix Jxt that is associated with Cartesian space.The stabilized discrete equations in the diffusion form can be expressed in Cartesianspace:∫
VE
NirdV −∫
ΓE
Ni(n · a)−(ψn+θ − ψn+θbc )dΓ +
∫VE
(∇Ni)Tν∇ψn+1dV = 0, (24)
or in a form where we apply integration by parts of the transport terms once:∫VE
Ni(at(ψn+1 − ψn
∆t) + σψn+θ − sn+θ)dV −
∫VE
∇ · (Nia)ψn+θdV
+
∫ΓE
Ni(n · a)−(ψn+θbc )dΓ +
∫ΓE
Ni(n · a)+ψn+θdΓ +
∫VE
(∇Ni)Tν∇ψn+1dV = 0. (25)
3. Coupled Equations
3.1. Non-Linear Petrov-Galerkin coupled equations
The Petrov-Galerkin method discussed above is further applied to the time dependentcoupled transport equations:
Axt · ∇xtΨ = s, (26)
where for 1D Axt = (At Ax)T and in 3D Axt = (At Ax Ay Az)T and in 1D thisequation becomes:
At∂Ψ
∂t+ Ax
∂Ψ
∂x= s. (27)
For coupled equations the projection of Axt onto ∇xtΨ can be written:
A∗xt = V(Axt · ∇xtΨ)V(||∇xtΨ||22)−1∇xtΨ. (28)
Thus
A∗xt · ∇xtΨ = Axt · ∇xtΨ, (29)
or (V(Axt · ∇xtΨ)V(||∇xtΨ||
22)−1∇xtΨ
)· ∇xtΨ = Axt · ∇xtΨ, (30)
6
where V(g) is a diagonal matrix in which V(g)µµ = gµ and the vector ||∇xtΨ||22 is such
that the µth entry is ||∇xtΨ||22µ = (∇xtΨµ) ·(∇xtΨµ). Since the matrix A∗xt has a block diag-
onal structure the transport equations (26) are a set ofM independent scalar equationsin which the µth scalar equation is expressed as :
a∗t µ∂Ψµ
∂t+ a∗xµ
∂Ψµ
∂x+ a∗yµ
∂Ψµ
∂y+ a∗zµ
∂Ψµ
∂z= sµ, (31)
and a∗t µ = A∗t µµ, a∗xµ = A∗xµµ, a∗yµ = A∗yµµ, a∗zµ = A∗zµµ. Since the equations have beenuncoupled then the scalar equation methods described in the previous section can nowbe applied. This is effectively done below.The Petrov-Galerkin’s modified form of the differential equation is:
(I − (∇xt · A∗xt)T P∗xt)(Axt · ∇xtΨ − s) = 0. (32)
Testing equation (32) with a diagonal matrix of space-time basis function Nxti (this hasthe basis function Nxti along its main diagonal), integrating over a single element VE
and applying integration by parts results in:∫VE
NxtirdV −∫
ΓE
Nxti(nxt · Axt)−(Ψ − Ψbc)dΓ
+
∫VE
((∇xtNxti) · A∗xt)T P∗xtrdV +
∫ΓE
Nxtinxt · A∗xtP∗xtrdΓ = 0, (33)
with a finite element expansion Ψ =∑N
j=1 Nxt jΨ j (where Ψ j is the orderM vector ofunknowns at node j) and r = Axt · ∇xtΨ − s.Using the eigen-decomposition nxt ·Axt = LxtΛxtRxt then (nxt ·Axt)− = LxtΛ
−xtRxt with
Λ−xtkk = min0,Λxtkk, where Λxt is the diagonal matrix whose diagonal elements arethe corresponding eigenvalues, and
AxtRxt = LxtΛxt; ATxtLxt = RxtΛxt, (34)
where the matrices Lxt and Rxt, whose columns are the normalized singular vectors,satisfy LT
xtLxt = I and RTxtRxt = I. This eigen decomposition enables the boundary
condition to be applied to incoming information only. We apply a zero boundary con-dition for the residual r = 0 which results in:∫
VE
NxtirdV −∫
ΓE
Nxti(nxt ·Axt)−(Ψ−Ψbc)dΓ+
∫VE
((∇xtNxti) ·A∗xt)T P∗xtrdV = 0. (35)
P∗xt is a function of A∗xt and the size and shape of the elements, for example:
P∗xt =14
(|A∗xt · ∇xtNxti|)−1, (36)
or using the space-time Jacobian matrix Jxt:
P∗xt =14
(||J−1xt A∗xt ||2)−1. (37)
7
Since the matrices A∗t , A∗x, A∗y, A∗z that go into constructing A∗xt = (A∗tT ,A∗x
T ,A∗yT ,A∗z
T )T
are diagonal the matrix P∗xt is also diagonal. In the traditional Petrov Galerkin methodA∗xt = Axt in the above and Pxt replaces P∗xt. In a similar way to the scalar equation,the value of P∗xt can be adjusted to ensure that the resulting value of P∗xt is not so largethat there is more transport backwards than forwards in the resulting discrete system ofequations by:
P∗xt = minE−1,14
(||J−1xt A∗xt ||2)−1. (38)
where the diagonal entries of the matrix E are positive and E contains small positivenumbers (tolerances) used to avoid a ”division by zero or near zero” error, e.g. 1×10−10.In the above we can work with the stabilization in the diffusion form with:
∫VE
NxtirdV −∫
ΓE
Nxti(nxt · Axt)−(Ψ − Ψbc)dΓ +
∫VE
(∇xtNxti)T K∇xtΨdV = 0, (39)
or in a form where we apply integration by parts of the transport terms once:∫VE
Nxti(−s)dV −∫
VE
(∇xt · (NxtiAxt))ΨdV
+
∫ΓE
Nxti(nxt · Axt)−ΨbcdΓ +
∫ΓE
Nxti(nxt · Axt)+ΨdΓ +
∫VE
(∇xtNxti)T K∇xtΨdV = 0,
(40)in which theM×M diagonal matrix containing the diffusion coefficients is:
K = V(Axt · ∇xtΨ)P∗xtV(||∇xtΨ||22)−1V(r). (41)
The resulting diagonal matrix K can be modified to ensure non-negative diffusion bysetting any of its negative entries to zero or taking their absolute values. Alternativelyone can work with the residual only, by replacing Axt · ∇xtΨ with the residual r, whichresults in:
K = V(r)T P∗xtV(||∇xtΨ||22)−1V(r), (42)
which is always positive because P∗xt is positive semi-definite (as well as diagonal) andin which V(r) is the diagonal matrix containing the residual of the governing equationson its diagonal. Equation (42) for the diffusivity can be derived by re-defining A∗xt inequation (28) to:
A∗xt = V(r)V(||∇xtΨ||22)−1∇xtΨ. (43)
3.2. Simplified coupled equations
Assuming time is discretised using the two level θ-method:
r = AtΨn+1 − Ψn
∆t+ A · ∇Ψn+θ − sn+θ, (44)
8
with A = (Ax Ay Az)T and Ψn+θ = ΘΨn+1 + (I−Θ)Ψn in which Θ is a diagonal matrixcontaining the time stepping parameters and also defining
∇xtΨ = (Ψn+1 − Ψn
∆t, (∇Ψn+θ)T )T . (45)
Using this definition equation (45) enables the application of the formalism of space-time discretisation, developed here, for example:
A∗xt = (A∗tT , A∗T )T = V(Axt · ∇xtΨ)V(||∇xtΨ||
22)−1∇xtΨ (46)
and
P∗xt = minE−1, (14
(||J−1A∗||2)−1. (47)
By applying diffusion only in Cartesian space the stabilized discrete equations in dif-fusion form can be written:∫
VE
NirdV −∫
ΓE
Ni(n · A)−(Ψn+θ − Ψn+θbc )dΓ +
∫VE
(∇Ni)T K∇Ψn+1dV = 0, (48)
or in a form where we apply integration by parts once to the transport terms:∫VE
Ni(AtΨn+1 − Ψn
∆t− sn+θ)dV −
∫VE
∇ · (NiA)Ψn+θdV
+
∫ΓE
Ni(n · A)−Ψn+θbc +
∫ΓE
Ni(n · A)+Ψn+θdΓ +
∫VE
(∇Ni)T K∇Ψn+1dV = 0. (49)
4. Stable Reduced Order Modelling using Diffusion from Petrov-Galerkin Meth-ods
In this section, the Petrov-Galerkin method discussed above is applied to form conser-vative stablisation methods for ROM’s which for non-linear problems have a tendencyto diverge due to inadequate sub-grid-scale modelling (if Galerkin methods are appliede.g. the POD method).
4.1. Derivation of Petrov-Galerkin POD modelsIf the original discrete system at a given time level is
AΨ = b, (50)
its modified discrete system is then written:
CT W−1AΨ = CT W−1b, (51)
in which for least squares, C = A. Note that the solution of this equation (51) isthe same as the original system (50) however critically, it is not so when the reduced
9
order modelling is applied. The weighting matrix W can be chosen such as to renderthe system of equations dimensionally consistent (and thus may contain characteristicdimensions such as the time step size ∆t and a length scale) and also contain the massmatrix of the system. Least Squares (LS) methods have dissipative properties, unlikeGalerkin methods, but are not generally conservative for coupled systems of equations.However LS methods may be applied at each equation level to make them conservative,and in which case C may contain just parts of the matrix A.Using POD methods any variable ψ can be approximately expressed as an expansionof the first few POD basis functions φ1, . . . , φM:
ψ(x, t) = ψ(x) +
M∑m=1
αm(t)φm(x), (52)
where ψ is the mean of the variables ψ over the time, αm (1 ≤ m ≤ M) are the time-dependent coefficients to be determined.In FE method, the variable ψ can be expressed as:
ψm =
N∑j=1
Nxt jΨm j, (53)
and the POD basis functions:
φm =
N∑j=1
Nxt jΦm j. (54)
Thus, taking into account (52), the variables Ψ = (Ψ1, . . . ,ΨN ) can be written:
Ψ = Ψ + MPODΨPOD, (55)
where ΨPOD = (α1, . . . , αM)T , and the matrix MPOD which consists of the POD basisfunctions, i.e. MPOD = [Φ1, . . . ,ΦM] ∈ RN×M , in which Φm = (Φm 1, . . . ,ΦmN )T
(1 ≤ m ≤ M).The reduced order model can be obtained by projecting the original models onto thereduced space, i.e. in a method analogous to the Galerkin method, taking the PODbasis function as the test function, then integrating it over the computational domainΩ. By multiplying (50) by the matrix MPODT and taking into account (55), the PODdiscrete model of (50) is obtained:
MPODT AMPODΨPOD = MPODT (b − AΨ). (56)
For the LS method (see (51)), equation (56) is:
MPODT CT W−1AMPODΨPOD = MPODT CT W−1(b − AΨ). (57)
Using the non-linear Petrov-Galerkin mechanics discussed in section 3, the POD re-duced order model can thus be written:
MPODT (I + CT W−1)AMPODΨPOD = MPODT (I + CT W−1)(b − AΨ), (58)
10
or in diffusion form (analogous to (39) - (41)):
(MPODT AMPOD + K)ΨPOD = MPODT (b − AΨ), (59)
where the calculation of the diffusion matrix K will be discussed below in sections 4.2and 5. As we know, a common solution to the divergence of ROM solutions (56) is toadd diffusion terms into the equations and tune these diffusion terms to a best matchthe full model solution. The nonlinear Petrov-Galerkin POD method (59) presents anatural and easy way to introduce a diffusion term into ROM without tuning/optimisingand provides appropriate modeling and stabilisations for the numerical solution of highorder nonlinear PDEs.
4.2. Calculation of the diffusion coefficientReminding that the construction of the POD model is similar to that of the Galerkin FEmodel, the diffusion coefficient in K can thus be calculated (analogous to (17)):
νPOD = κrPOD p∗xt
PODrPOD
||∇xtψPOD||22
, (60)
where
p∗xtPOD
= mink14
∣∣∣∣∣∣ a∗xt · ∇xtNxtPODk
maxx|NxtPODk (x)|
∣∣∣∣∣∣−1
, (61)
in which we have assumed that the length scale associated with the POD method is thegradient of the normalised POD basis function in the direction a∗xt, and κ is a scalar (the-oretically it is 1.0, numerically less than 1.0). The residual vector can be determinedfrom:
rPOD = EPOD−1((MPODT AMPOD)ΨPOD −MPODT (b − AΨ)), (62)
where EPODi j =
∫V ΨPOD
i ΨPODj dV , taking into account the ROM basis functions are
orthonormal, so EPOD = I. Since the absolute value of the residual may not matter:
rPOD = EPOD−1MPODT|AMPODΨPOD − (b − AΨ)|, (63)
Assuming νPOD is represented using a POD expansion (νPOD =∑M
k=1PODNPOD
k νPODk ),
the vector νPOD = (νPOD1 , . . . , νPOD
M ) is thus written (details in Appendix A):
νPOD = |14
OPOD−1UPOD−1q|, (64)
and ∫V
NxtPODm qdV =
∫V
NxtPODm rPODrPODdV. (65)
Using q =∑M
k=1 NxtPODk qk and the orthonormal property of POD the mth equation of
equation (65) becomes:
qm =
∫V
NxtPODm rPODrPODdV, (66)
11
with q = (q1 q2 ... qNPOD )T , and
UPODmk =
∫V
NxtPODm ((
∂ψ
∂t)2 + (
∂ψ
∂x)2 + (
∂ψ
∂y)2 + (
∂ψ
∂z)2 + ε)Nxt
PODk dV, (67)
in which the small scalar ε ensures that U is non-singular. The element of the matrixOPOD in (64) is
OPODmk =
∫V
NxtPODm maxnε, |a∗xt · ∇xtNxtn|Nxt
PODk dV = (MPODT OMPOD)mk, (68)
where Oi j =∫
V Nxtimaxnε, |a∗xt · ∇xtNxtn|Nxt jdV . Using the eigen-decompositionBPOD = LPOD
B ΛPODB RPOD
B , where
Bi j =
∫V
NxtPODi a∗xt · ∇xtNxt
PODj dV, (69)
then the matrix OPOD−1 in (64) can be written:
OPOD−1= RBPOD |ΛBPOD |−1LBPOD . (70)
An alternative definition to OPOD−1 defined by equation (70) is:
OPOD−1= |BPODT BPOD|−
12 = LBPODT BPOD |ΛBPODT BPOD |
− 12 RBPODT BPOD . (71)
Thus |ΛBPODT BPOD |− 1
2 should have similar values to p∗xt calculated from equation (61). Agood choice suggested by the non-linear Petrov-Galerkin approach for C in equation(58) is C = B which is very similar to the diffusion based method.
5. Bassi Rebay representation of DG diffusion
One of the key ingredients of DG methods is the formulation of interface (numerical)fluxes, which provide a weak coupling between the unknowns in neighboring elements.The classic approach introduced by Bassi and Rebay in 1997 [8] is here used for therepresentation of the DG diffusion term. The diffusion term is written:∑
ele
∫VE
Ni∂
∂xν∂ψ
∂xdVE = −
∑ele
∫VE
ν∂Ni
∂x∂ψ
∂xdVE +
∑ele
∫ΓE
νNi nx ·∂ψ
∂xdΓE . (72)
Defining
ψx =∂ψ
∂x, (73)
the diffusion term (72) can be rewritten:∑ele
∫VE
∂
∂xνNi
∂ψ
∂xdVE = −
∑ele
∫VE
ν∂Ni
∂xψxdVE +
∑ele
∫ΓE
νNi nx · ψxdΓE
= −∑ele
∫VE
ν∂Ni
∂xψxdVE +
∑ele
∫ΓE
νNi nx ·12
(ψxcur + ψxnei)dΓE
= CTψx, (74)
12
where ψxcur and ψxnei represent ψx at the current and neighbouring elements respec-tively, and ψx is zero on the boundary, i.e. zero through the surface integral. Multiply-ing (73) by Ni and integrating it over the domain V , obtains∑
ele
∫VE
NiψxdVE =∑ele
∫VE
Ni∂ψ
∂xdVE
= −∑ele
∫VE
∂Ni
∂xψdVE +
∑ele
∫ΓE
Ni nx · ψdΓE
= −∑ele
∫VE
∂Ni
∂xψdVE +
∑ele
∫ΓE
Ni nx ·12
(ψcur + ψnei)dΓE
=∑ele
∫VE
Ni∂ψ
∂xdVE −
∑ele
∫ΓE
Ni nx · ψcurdΓE +∑ele
∫ΓE
Ni nx ·12
(ψcur + ψnei)dΓE
=∑ele
∫VE
Ni∂ψ
∂xdVE −
∑ele
∫ΓE
Ni nx ·12
(ψcur − ψnei)dΓE
= FΨ, (75)
where ψcur and ψnei represent ψ at the current and neighbouring elements respectively.Equation (75) can be rewritten
Mψx = FΨ. (76)
where M is the mass matrix in FE, and ψx =∂ψ∂x can be calculated
ψx = M−1FΨ. (77)
The diffusion term (74) can be-rewritten:∑ele
∫VE
Ni∂
∂xν∂ψ
∂xdVE = CT M−1FΨ. (78)
The approach discussed above is easily extended to 2D and 3D cases.
6. Example cases: advection and shock waves
Shock waves are characterized by an abrupt, nearly discontinuous change in the char-acteristics of the medium. Across a shock there is always an extremely rapid rise inthe flow quantities (e.g. density, velocity, pressure, entropy) [21]. The shock waves aredescribed by the nonlinear hyperbolic Euler equations in a non-conservation form:
∂ψ
∂t+ u
∂ψ
∂x= 0. (79)
In this work, the Petrov-Galerkin POD method discussed above is applied to (79). Adiffusional DG is used to control numerical oscillations. The root mean square error(RMSE) and correlation coefficient of results between the POD and full models at thetime level n is used to estimate the error of the POD/ROM projection results:
RMS En =
√∑Ni=1(ψn
i − ψno,i)
2
N, (80)
13
where, ψni and ψn
o,i are the vectors containing the POD and full model results at the nodei respectively. The correlation coefficient of results between the POD and full modelsat the time level n with expected values µψn and µψn
o and standard deviations σψn andσψn
o is defined as:
corr(ψn, ψno)n =
cov(ψn, ψno)
σψnσψno
=E[(ψn − σψn )(ψn
o − σψno )
σψnσψno
. (81)
6.1. Case 1: 1D advectionThe problem solved here has a unit 1D domain and open boundary conditions at bothends and space and time steps of ∆t = 0.001, ∆x = ∆y = 0.005 respectively. Thesimulation period is [0, 0.2]. The initial condition is
ψt=0 = 1.0 0.2 < x < 0.4;
ψt=0 = 0.0 x ≤ 0.2; x ≥ 0.4. (82)
Figure 1 shows the comparison of the results between the full and POD models. It isobserved that the instability in the POD results is reduced by using the Petrov-GalerkinPOD methods (figure 1). The RMSE and correlation coefficient of results betweenthe POD and full models during the simulation period are provided in figure 2. It canbe seen that by using the Petrov-Galerkin POD model, the RMSE of results betweenthe POD and full models is reduced up to 20% of that of the Galerkin POD modelingat the end of the simulation period (with 10 POD bases, where 88% of the originalenergy can be represented). The correlation coefficient of results mostly larger than98% when the Petrov-Galerkin method is used. With an increase of the number ofPOD bases (where 99% of the original energy can be captured with 20 POD bases) theRMSE of POD results using the Petrov-Galerkin method is further reduced and attainsa small value (less than 0.02) while the correlation coefficient achieves 99.9%. Thescalar diffusion coefficient used in the Petrov-Galerkin POD model can be calculatedby either (60) or (64) and the corresponding results are shown in figures 3 and 4. ThePetrov-Galerkin POD results using equation (64) are much smoother than those withthe use of (60). The Petrov-Galerkin POD model is also applied to equation (79) withthe initial condition:
ψt=0 = e−(x−0.3)2
0.12 . (83)
The results from the POD model exhibit an overall good agreement with those obtainedwith the full model (figure 5) with 5 POD bases.
6.2. Case 2: 2D advectionThe Petrov-Galerkin POD model in conjunction with the Bassi-Rebay approach is ap-plied to a 2D advection case. The problem solved here has a unit square domain withopen boundary conditions and space and time steps of ∆t = 0.001, ∆x = ∆y = 0.01.The simulation period is [0, 0.2]. The initial condition is
ψt=0 = 1.0√
(x − 0.5)2 + (y − 0.5)2 ≤ 0.05;
14
(a) 10 POD bases (b) 10 POD bases
(c) 20 POD bases (d) 20 POD bases
Figure 1: Case 1 (1D advection): Comparison of the results between the full and POD models. Top panel:10 POD bases which represent 88% of the original energy; Bottom panel: 20 POD bases which represent99% of the original energy.
15
(a) RMSE (10 POD bases) (b) RMSE (20 POD bases)
(c) Correlation (10 POD bases) (d) Correlation (20 POD bases)
Figure 2: Case 1 (1D advection): RMSE and correlation coefficient of results between the POD and fullmodels (Left: 10 POD bases are used to represent 88% of the original energy; Right: 20 POD bases are usedto represent 99% of the original energy; where κ is the scalar in (60)).
16
(a) 10 POD bases (b) 10 POD bases
(c) 20 POD bases (d) 20 POD bases
Figure 3: Case 1 (1D advection): Comparison of results obtained from the Petrov-Galerkin POD modelusing: the first approach (60 and κ = 1.0) (; and the second approach (64 and κ = 0.01) used with (71.
17
(a) RMSE (10 POD bases) (b) RMSE (20 POD bases)
(c) Correlation (10 POD bases) (d) Correlation (20 POD bases)
Figure 4: Case 1 (1D advection): RMSE and correlation coefficient of results between the full and Petrov-Galerkin POD models using the first approach (60); and the second approach (64) used with (71).
18
(a) 2 POD bases (b) 5 POD bases
Figure 5: Case 1 (1D Gaussian wave): Comparison of results between the full and POD models (wherethe first approach (60); and the second approach (64) used with (71), 5 POD bases represent 98.4% of theoriginal energy while 2 POD bases represent 80% of the original energy).
ψt=0 = 0.0√
(x − 0.5)2 + (y − 0.5)2 > 0.05. (84)
A comparison of the results between the POD and full model is displayed in figures 6and 7. It can be seen that the POD reduced order results using the Galerkin approachbecome oscillatory and unstable and the RMSE of results increases as the simulationtime increases. By using the Petrov-Galerkin POD approach, the RMSE of results isreduced by a factor of 2 during the period [0.05s, 0.2s] while the correlation coefficientincreases from 99.8% to 99.95%.
6.3. Case 3: the Sod shock tube problem
The Petrov-Galerkin POD model is further evaluated in the classic Sod shock tubeproblem [19]. The POD model results are compared with the analytical solution [19]and those from the full model. The problem solved here has a unit 1D domain and openboundary conditions at both ends and space and time steps of ∆t = 0.001, ∆x = 0.0238.The simulation period is [0, 0.2s]. The Euler equation (79) can be re-written for an idealgas [20, 21]: ρ
ρue
t
+
ρuρu2 + pu(e + p)
x
=
000
, (85)
where ρ is the density of the fluid, u is the fluid velocity, e is the energy per unit volume(length), and p is the pressure. The equation of state is
p = (γ − 1)(e −
12ρu2
)(86)
19
(a) Results from the Galerkin POD model at t = 199 s (b) Results from the Petrov-Galerkin POD model at t = 199 s
(c) Results from the full model at t = 199 s
Figure 6: Case 2 (2D 45o advection): Comparison of the results obtained from the POD models and fullmodel, where 15 POD bases are used, which represent 95% of the original energy.
20
(a) RMSE (10 POD base) (b) Correlation (10 POD bases)
Figure 7: Case 2 (2D 45o advection):.RMSE and correlation coefficient of tracer results between the PODand the full models
where γ is the adiabatic gas index. For an ideal gas γ = 1.4. The test consists of a 1DRiemann problem with the following parameters, for left and right states of an idealgas: ρL
pL
uL
=
0.1250.10.0
, ρR
pR
uR
=
1.01.00.0
. (87)
The results from the POD and full models are provided in figure 8. It is shown (figure 8)that the Petrov-Galerkin POD model does well in capturing and resolving discontinu-ities of shock waves. The results reproduce the correct density and velocity profiles ofthe rarefaction wave. When 15 POD bases are used (where 95% of the original energyis captured), the RMSE of results between the Petrov-Galerkin POD and full modelsis less than 0.0075 and is further reduced to 0.001 with 25 POD bases (where 99.4%of the original energy is represented). The correlation coefficient of results achieves avalue of 99.99%.
7. Conclusions
A new Petrov-Galerkin method for stablisation of high order nonlinearity in reducedorder modelling is developed. The POD discontinuous Galerkin (DG) reduced ordermodel is applied to 1D and 2D advection, and 1D shock wave cases. A comparison ofthe results between the POD models using the Petrov-Galerkin and traditional Galerkinapproaches is carried out. It is observed that the instability in the POD results is reducedby using the Petrov-Galerkin POD methods which naturally introduces a diffusion terminto ROM. The RMSE of results between the POD and full models is decreased by20% − 60% while the correlation coefficient is mostly larger than 99.5% when thePetrov-Galerkin approach is used in conjunction with ROM. The Petrov-Galerkin POD
21
(a) Density (b) Density
(c) Velocity (d) Velocity
Figure 8: Case 3 (the Sod shock tube problem): Comparison of the results obtained from the POD and fullmodel at time level t = 200 s (where 15 POD bases are used to represent 95% of the original energy).
22
(a) Density (b) Density
(c) Velocity (d) Velocity
Figure 9: Case 3 (the Sod shock tube problem): Comparison of the results obtained from the POD and fullmodel at time level t = 200 s (where 25 POD bases are used to represent 99.4% of the original energy).
23
(a) RMSE (15 POD bases) (b) RMSE (25 POD bases)
(c) Correlation (15 POD bases) (d) Correlation (25 POD bases)
Figure 10: Case 3 (the Sod shock tube problem):.RMSE of results between the POD model and the fullmodel (where 15 POD bases are used to represent 95% of the original energy while 25 POD bases represent99.4% of the original energy).
24
model does well in capturing and resolving discontinuities of shock waves. The resultsobtained reproduce the correct density and velocity profiles of the rarefaction wave.
Future work will investigate impact of DEIM methodology on additional economyin CPU time (due to nonlinear effects) and apply this new Petrov-Galerkin POD ap-proach to more complex fluid flow models and optimal control of shock waves for theSod problem [21]. We envisage to compare in future research work the new approachwith the calibration method, e.g. Noack’s method [22] and the method proposed byZhu et al(2011) [23].
Acknowledgments
This work was carried out under funding from the UK’s Natural Environment ResearchCouncil (projects NER/A/S/2003/00595, NE/C52101X/1 and NE/C51829X/1), the En-gineering and Physical Sciences Research Council (GR/R60898 and EP/I00405X/1),and with support from the Imperial College High Performance Computing Service.Prof. I. M. Navon acknowledges the support of NSF/CMG grant ATM-0931198. Dr.Juan Du acknowledges the support of National Natural Science Foundation (Grant No.41075064) and National Basic Research Program of China (Grant No. 2012CB417404),and support from Prof. Zhu.
Appendix A. Derivation of POD DG diffusion coefficient
The Petrov-Galerkin’s modified form of the differential equation (32) is re-written here:
(I − (∇xt · A∗xt)T P∗xt)(Axt · ∇xtΨ − s) = 0. (A.1)
Multiplying (A.1) by a space-time basis function Nxti and integrating by parts over asingle element VE (here DG methods are employed), a general form of the Petrov-Galerkin operator for finite element methods is:∫
VE
Nxti(I +
(A∗xt · ∇
)P∗xt
)r dV. (A.2)
The finite element Riemann method is defined with
P∗xt =h2|nxA∗x + nyA∗y + nzA∗z |
−1, (A.3)
where h is the length size of the element. This is the approach used with Riemannsolvers that are implemented with a control volume discretisation in which case (nx, ny, nz)is the direction normal to a control volume. For finite elements nx = Ni x/n, ny = Niy/nand nz = Niz/n where n∗ is the normalisation factor. It should be noted that the trans-port term pre-multiplied by I when integrated by parts is conservative and has much incommon with control volume methods. Using the basis functions to define the lengthscale h
2 associated with the basis functions one can eliminate out the normalizationfactor to obtain:
P∗xt = |Nxti xA∗x + NxtiyA∗y + NxtizAxt∗z |−1 = |A∗xt · ∇Nxti|
−1. (A.4)
25
This is important as applying this method to reduced order basis functions there is nowell defined length scale except through this gradient of the basis functions.Since the POD matrices may be relatively small one can manipulate their eigen struc-ture in order to construct stabilization methods. For example, defining
BPODi j =
∫VE
NxtPODi A∗xt · ∇Nxt
PODj dV, (A.5)
then ∫VE
((∇xtNxtPODi ) · A∗xt)
T P∗xtrdV = (BPODT OPOD−1lPOD)i, (A.6)
withlPODi =
∫VE
NxtPODi rdV, (A.7)
OPOD−1= |BPOD|−1 = RBPOD |ΛBPOD |−1LBPOD . (A.8)
RBPOD , LBPOD are matrices of right and left eigen-vectors of BPOD and ΛBPOD is thematrix of eigen-values of BPOD. A possible alternative to OPOD−1 which would ensurepositive diffusion is:
OPOD−1= (BPODT BPOD)−
12 . (A.9)
Now since (BPOD)i is analogous to (∇xtNxtPODi ) · A∗xt of (A.6), (lPOD)i is analogous to
r of (A.6), and OPOD−1 is analogous to P∗xt they replace them in the definition of thediffusion coefficient in equation (42) which then results in equation (64).
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