Non-linear Petrov–Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods

doi:10.1016/j.jcp.2012.10.011

Journal of Computational Physics

Volume 234, 1 February 2013, Pages 540-559

https://doi.org/10.1016/j.jcp.2012.10.011 Get rights and content

Abstract

A new Petrov–Galerkin approach for dealing with sharp or abrupt field changes in discontinuous 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 ROM without tuning/optimising and provides appropriate modelling and stablisation for the numerical solution of high order nonlinear PDEs. The approach is based 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 stabilization of the proper orthogonal decomposition (POD) model using the new Petrov–Galerkin approach is demonstrated in 1D and 2D advection and 1D shock wave cases. Error estimation is carried out for evaluating the accuracy of the Petrov–Galerkin POD model. Numerical results show the new nonlinear Petrov–Galerkin method is a promising approach for stablisation of reduced order modelling.

Introduction

Reduced order model (ROM) technology is a rapidly growing discipline, with significant potential advantages in: interactive use, emergency response, ensemble calculations and data assimilation [1], [2], [3], [4]. ROM is expected to play a major role in facilitating real-time turn-around with computational results and data assimilation. Most model reduction methods can be viewed as approximation methods by projection (for comprehensive description see [5], [6]). Most of those methods (e.g., balanced truncation) are designed for stable, linear and moderate-order systems (state orders are less than $10^{4}$ ) [7], so are not practical for many fluids systems although they can provide accurate low-order representations of state-space systems. Among existing approaches, the proper orthogonal decomposition (POD) method provides an efficient means of deriving the reduced basis for nonlinear partial differential equations (PDEs). Using the POD technique, it is possible to extract a set of modes characteristic of the database which constitutes the optimal basis for the energetic description of the flow. A Galerkin projection of the original equations onto a finite number of POD bases yields a set of ordinary differential equations in time. However, due to the energetic optimality of the POD bases, only few modes are sufficient to give a good representation of the kinetic energy of the flow. The leading POD modes are not able to dissipate enough energy since the main amount of viscous dissipation takes place in the small eddies (unresolved modes) [9]. Galerkin POD methods may thus suffer from a lack of numerical stability especially for high order nonlinear PDEs. There are various ways to recover the effect of the truncated bases (usually the small scales, i.e. unresolved modes) and improve the numerical 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, streamline diffusion) into the POD reduced order equations [12], [13];
3.
the approach of Noack et al. [14], that uses a finite-time thermodynamics formalism;
4.
residual based stablisation approach [9] where the unsolved terms are represented by a number of residual modes which are calculated by the residuals of ROMs.

For efficient calculation of nonlinear terms, the discrete empirical interpolation method (DEIM) [15] provides a dimension reduction of the nonlinear term by replacing the non-linear terms with a coefficient-function approximation consisting of a linear combination of pre-computed basis functions and parameter-dependent coefficients.

However the main drawback of the above stabilization methods is that there are always some parameters to tune/optimise for a best match the full solution. More recently, the Petrov–Galerkin method has been introduced to POD and applied to the 1D non-linear static problem and ODE [16]. This method presents a natural and easy way to introduce a diffusion term into ROM without tuning/optimising and provides appropriate modelling 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 discontinuous Galerkin modelling in order to control numerical oscillations. The approach is based 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 organised as follows. Section 2 provides the derivation of the new Petrov–Galerkin approach for one scalar time dependent transport equation. This is then followed by the extension to coupled time dependent equations and an example is provided by two-time level discrete equations in Section 3. Section 4 addresses the issue of how stable reduced order modelling is performing using the new Petrov–Galerkin approach. In Section 5, a Bassi Rebay representation of discontinuous Galerkin methods for the diffusion term is described. The method is applied to 1D and 2D advection and shock cases in Section 6 followed by discussion of the numerical results. Finally, conclusions are drawn in Section 7.

Section snippets

Non-linear Petrov–Galerkin scalar equation

The one scalar time dependent transport equation assumes the form: $a_{xt} \cdot \nabla_{xt} ψ = s,$ where $ψ$ represents field states (e.g. temperature, pollutants); s is the source term; $a_{xt} = (a_{t} a)^{T}$ and $a = (a_{x} a_{y} a_{z})^{T}$ (here, the velocity vector); $a_{t}, a_{x}, a_{y}, a_{z}$ are the coefficients of the time and space derivatives along the x, y, z direction respectively. For simplicity, this Eq. (1) in 1D with time dependence becomes: $a_{t} \frac{\partial ψ}{\partial t} + a_{x} \frac{\partial ψ}{\partial x} = s .$ Using the cosine rule between the two vectors $a_{xt}$ and $\nabla_{xt} ψ$ : $\cos (θ_{a}) = \frac{a_{xt} \cdot \nabla_{xt} ψ}{| a_{xt} | | \nabla_{xt} ψ |},$

Non-linear Petrov–Galerkin coupled equations

The Petrov–Galerkin method discussed above is further applied to the time dependent coupled transport equations: $A_{xt} \cdot \nabla_{xt} Ψ = s,$ where for 1D $A_{xt} = (A_{t} A_{x})^{T}$ and in 3D $A_{xt} = (A_{t} A_{x} A_{y} A_{z})^{T}$ in which the matrices $A_{t}, A_{x}$ , $A_{y}$ and $A_{z}$ contain the coefficients of the derivatives of scalars with respect to time t, coordinates $x, y$ and z respectively. In 1D Eq. (26) becomes: $A_{t} \frac{\partial Ψ}{\partial t} + A_{x} \frac{\partial Ψ}{\partial x} = s .$ For coupled equations the projection of $A_{xt}$ onto $\nabla_{xt} Ψ$ can be written: $A_{xt}^{*} = V (A_{xt} \cdot \nabla_{xt} Ψ) V (‖ \nabla_{xt} Ψ ‖_{2}^{2})^{- 1} \nabla_{xt} Ψ .$ Thus $A_{xt}^{*} \cdot \nabla_{xt} Ψ = A_{xt} \cdot \nabla_{xt} Ψ,$ or $(V (A_{})$

Stable reduced order modelling using diffusion from Petrov–Galerkin methods

In this section, the Petrov–Galerkin method discussed above is applied to form conservative stablisation methods for ROM’s which for non-linear problems have a tendency to diverge due to inadequate sub-grid-scale modelling (if Galerkin methods are applied e.g. the POD method).

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 neighbouring elements. The classic approach introduced by Bassi and Rebay in 1997 [8] is here used for the representation of the DG diffusion term. The diffusion term is written: $\sum_{ele} \int_{V_{E}} N_{i} \frac{\partial}{\partial x} ν \frac{\partial ψ}{\partial x} {dV}_{E} = - \sum_{ele} \int_{V_{E}} ν \frac{\partial N_{i}}{\partial x} \frac{\partial ψ}{\partial x} {dV}_{E} + \sum_{ele} \int_{Γ_{E}} ν N_{i} n_{x} \cdot \frac{\partial ψ}{\partial x} d Γ_{E} .$ Defining $ψ_{x} = \frac{\partial ψ}{\partial x},$ the diffusion term (73) can be rewritten: $\sum_{ele} \int_{V_{E}} \frac{\partial}{\partial x} ν N_{i} \frac{\partial ψ}{\partial x} {dV}_{E} = - \sum_{ele} \int_{V_{E}} ν \frac{\partial N_{i}}{\partial x} ψ_{x} {dV}_{E} + \sum_{ele} \int_{Γ_{E}} ν N_{i} n_{x} \cdot$

Example cases: advection and shock waves

Shock waves are characterised by an abrupt, discontinuous change in the characteristics of the medium. The shock waves are described by the nonlinear hyperbolic Euler equations in a non-conservation form: $\frac{\partial ψ}{\partial t} + a_{x} \frac{\partial ψ}{\partial x} = 0 .$ In this work, the Petrov–Galerkin POD method discussed above is applied to (80). A diffusional DG is used to control numerical oscillations. The root mean square error (RMSE), relative error (RE) and correlation coefficient of results between the POD and full models at the time

Conclusions

A new Petrov–Galerkin method for stablisation of high order nonlinearity in reduced order modelling is developed. The POD discontinuous Galerkin (DG) reduced order model is applied to 1D and 2D advection, and 1D shock wave cases. A comparison of the results between the POD models using the Petrov–Galerkin and traditional Galerkin approaches is carried out. It is observed that the instability in the POD results is reduced by using the Petrov–Galerkin POD methods which naturally introduces a

Acknowledgments

This work was carried out under funding from the UK’s Natural Environment Research Council (Projects NER/A/S/2003/00595, NE/C52101X/1 and NE/C51829X/1), the Engineering 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

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Non-linear Petrov–Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods

Abstract

Introduction

Section snippets

Non-linear Petrov–Galerkin scalar equation

Non-linear Petrov–Galerkin coupled equations

Stable reduced order modelling using diffusion from Petrov–Galerkin methods

Bassi Rebay representation of DG diffusion

Example cases: advection and shock waves

Conclusions

Acknowledgments

J. Comput. Phys.

J. Comput. Phys.

Ocean Model.

J. Comput. Phys.

Ann. Nucl. Energy

J. Comput. Phys.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Reduced order modelling of an adaptive mesh ocean model

Int. J. Numer. Methods Fluids

Efficiency of reduced-order, time-dependent adjoint data assimilation approaches

J. Oceanogr.

Model-reduced variational data assimilation

Mon. Wea. Rev.

Reduced-order 4d-var: a preconditioner for the incremental 4d-var data assimilation method

Geophys. Res. Lett.

Approximation of Large-scale Dynamical Systems

A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices

IEEE Trans. Comput. Aided Des. Integr. Circuits Syst.