Reduced Basis Method for Nonlinear Explicit Finite ... - CiteSeerX

Reduced Basis Method for Nonlinear Explicit Finite Volume Approximations of Hyperbolic Evolution Equations Bernard Haasdonk* Westf¨alische Wilhelms-Universit¨at M¨unster, Germany [email protected] Mario Ohlberger Westf¨alische Wilhelms-Universit¨at M¨unster, Germany [email protected]

We address the task of model reduction for parametrized scalar hyperbolic or convection dominated parabolic evolution equations. These are problems which are characterized by a parameter vector µ ∈ P from some set of possible parameters P ⊂ Rp , and the evolution problem is to determine u(t, µ) ∈ L∞ (Ω) ∩ L2 (Ω) on an open bounded domain Ω ⊂ Rd and finite time interval t ∈ [0, T ], T > 0 such that ∂t u(µ) + L(t, µ)(u(t, µ)) = 0,

u(0, µ) = u0 (µ),

and suitable boundary conditions are satisfied. Here u0 (µ) ∈ L∞ (Ω) ∩ L2 (Ω) are the parameter-dependent initial values, L(t, µ) is the parameter dependent spatial differential operator. We assume a discretization with explicit finite volume schemes and first order time discretization, which yields discrete solutions ukH (µ) ∈ WH , k = 0, . . . , K in the Hdimensional finite volume space WH ⊂ L∞ (Ω) ∩ L2 (Ω) approximating u(tk , µ) at the time instants 0 = t0 < t1 < . . . < tK = T . Such detailed simulations are frequently expensive to compute due to the space resolution and not suitable for use in multi-query settings, i.e. multiple simulation requests with varying parameters µ. Reduced Basis Methods are increasingly popular methods to solve such parametrized problems, aiming at a problemdependent simulation scheme, that approximates the detailed solutions ukH (µ) by efficiently computed reduced solutions ukN ∈ WN . Here WN ⊂ L2 (Ω) is an N -dimensional reduced basis space with suitable reduced basis ΦN which is generated in a problem specific way based on snapshots of detailed solutions for suitably chosen time instants ki and parameters µi ∈ P, i.e. ΦN ⊂ span{ukHi (µi )}. Reduced basis methods in particular have been applied successfully for various elliptic and parabolic problems, almost exclusively based on finite element discretizations. For linear elliptic problems we refer to [6], linear parabolic equations are treated in [2], extensions to nonlinear equations [8, 1] or systems [7] have been developed. Recently, we have proposed an RB-formulation for linear finite volume schemes [4] in case of so called affine parametric dependence of the data functions. We presented a method to construct reduced bases based on adaptive grids in parameter space [3]. We extended the finite volume RB-scheme to explicit discretizations with general parametric dependence and demonstrated the applicability to a linear evolution problem [5]. In the current presentation, we adopt the latter methodology to nonlinear equations. The key ingredient in the scheme is an empirical interpolation step for approximating the discrete spatial differential operator evaluations. The result is an approximating collateral reduced basis space WM for these operator-evaluations. As a sample problem for nonlinear hyperbolic equations, we consider the Burgers equation in 2D, which is parametrized via its initial conditions and velocity

field. An explicit finite volume scheme is applied as space discretization. By suitable, fully automatical, constructions of the reduced basis and collateral reduced basis, we obtain a reduced model, that enables rapid parameter variation with accurate approximations. Experimentally, we investigate the approximation properties and demonstrate the runtime gain compared to the full finite volume scheme.

References [1] M.A. Grepl. Reduced-basis Approximations and a Posteriori Error Estimation for Parabolic Partial Differential Equations. PhD thesis, Massachusetts Institute of Technology, May 2005. [2] M.A. Grepl and A.T. Patera. A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. M2AN Math. Model. Numer. Anal., 39(1):157–181, 2005. [3] B. Haasdonk and M. Ohlberger. Basis construction for reduced basis methods by adaptive parameter grids. In P. D´ıez and K. Runesson, editors, Proc. International Conference on Adaptive Modeling and Simulation, ADMOS 2007. CIMNE, Barcelona, 2007. [4] B. Haasdonk and M. Ohlberger. Reduced basis method for finite volume approximations of parametrized linear evolution equations. M2AN, Math. Model. Numer. Anal., 2008. Accepted. [5] B. Haasdonk, M. Ohlberger, and G. Rozza. A reduced basis method for evolution schemes with parameter-dependent explicit operators. Technical report, Preprint 09/07 - N, FB 10 , University of M¨unster, Preprint Nr. 17/2007, Institute of Mathematics, University of Freiburg, 2007. [6] A.T. Patera and G. Rozza. Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations . MIT, 2007. Version 1.0, Copyright MIT 2006-2007, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering. [7] G. Rozza. Shape design by optimal flow control and reduced basis techniques: Applications to bypass configurations in haemodynamics. PhD thesis, Ecole Polytechnique Federale de Lausanne, November 2005. [8] K. Veroy, C. Prud’homme, and A.T. Patera. Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds. C.R. Acad. Sci. Paris, Ser. I, 337:619–624, 2003.