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Chi-wang Shu - One of the best experts on this subject based on the ideXlab platform.
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an alternative formulation of finite difference weighted eno schemes with lax wendroff Time Discretization for conservation laws
SIAM Journal on Scientific Computing, 2013Co-Authors: Yan Jiang, Chi-wang Shu, Mengping ZhangAbstract:We develop an alternative formulation of conservative finite difference weighted essentially nonoscillatory (WENO) schemes to solve conservation laws. In this formulation, the WENO interpolation of the solution and its derivatives are used to directly construct the numerical flux, instead of the usual practice of reconstructing the flux functions. Even though this formulation is more expensive than the standard formulation, it does have several advantages. The first advantage is that arbitrary monotone fluxes can be used in this framework, while the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. The second advantage, which is fully explored in this paper, is that a narrower effective stencil is used compared with previous high order finite difference WENO schemes based on the reconstruction of flux functions, with a Lax--Wendroff Time Discretization. We will describe the scheme formulation and present numerical tests for one- and two-dimensional scalar ...
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high order conservative lagrangian schemes with lax wendroff type Time Discretization for the compressible euler equations
Journal of Computational Physics, 2009Co-Authors: Wei Liu, Juan Cheng, Chi-wang ShuAbstract:In this paper, we explore the Lax-Wendroff (LW) type Time Discretization as an alternative procedure to the high order Runge-Kutta Time Discretization adopted for the high order essentially non-oscillatory (ENO) Lagrangian schemes developed in [3,5]. The LW Time Discretization is based on a Taylor expansion in Time, coupled with a local Cauchy-Kowalewski procedure to utilize the partial differential equation (PDE) repeatedly to convert all Time derivatives to spatial derivatives, and then to discretize these spatial derivatives based on high order ENO reconstruction. Extensive numerical examples are presented, for both the second-order spatial Discretization using quadrilateral meshes [3] and third-order spatial Discretization using curvilinear meshes [5]. Comparing with the Runge-Kutta Time Discretization procedure, an advantage of the LW Time Discretization is the apparent saving in computational cost and memory requirement, at least for the two-dimensional Euler equations that we have used in the numerical tests.
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Efficient Time Discretization for local discontinuous Galerkin methods
Discrete & Continuous Dynamical Systems - B, 2007Co-Authors: Yinhua Xia, Chi-wang ShuAbstract:In this paper, we explore three efficient Time Discretization techniques for the local discontinuous Galerkin (LDG) methods to solve partial differential equations (PDEs) with higher order spatial derivatives. The main difficulty is the stiffness of the LDG spatial Discretization operator, which would require a unreasonably small Time step for an explicit local Time stepping method. We focus our discussion on the semi-implicit spectral deferred correction (SDC) method, and study its stability and accuracy when coupled with the LDG spatial Discretization. We also discuss two other Time Discretization techniques, namely the additive Runge-Kutta (ARK) method and the exponential Time differencing (ETD) method, coupled with the LDG spatial Discretization. A comparison is made among these three Time Discretization techniques, to conclude that all three methods are efficient when coupled with the LDG spatial Discretization for solving PDEs containing higher order spatial derivatives. In particular, the SDC method has the advantage of easy implementation for arbitrary order of accuracy, and the ARK method has the smallest CPU cost in our implementation.
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The discontinuous Galerkin method with Lax-Wendroff type Time Discretizations
Computer Methods in Applied Mechanics and Engineering, 2005Co-Authors: Jianxian Qiu, Michael Dumbser, Chi-wang ShuAbstract:Abstract In this paper we develop a Lax–Wendroff Time Discretization procedure for the discontinuous Galerkin method (LWDG) to solve hyperbolic conservation laws. This is an alternative method for Time Discretization to the popular total variation diminishing (TVD) Runge–Kutta Time Discretizations. The LWDG is a one step, explicit, high order finite element method. The limiter is performed once every Time step. As a result, LWDG is more compact than Runge–Kutta discontinuous Galerkin (RKDG) and the Lax–Wendroff Time Discretization procedure is more cost effective than the Runge–Kutta Time Discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics when nonlinear limiters are applied.
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Finite Difference WENO Schemes with Lax--Wendroff-Type Time Discretizations
SIAM Journal on Scientific Computing, 2003Co-Authors: Jianxian Qiu, Chi-wang ShuAbstract:In this paper we develop a Lax--Wendroff Time Discretization procedure for high order finite difference weighted essentially nonoscillatory schemes to solve hyperbolic conservation laws. This is an alternative method for Time Discretization to the popular TVD Runge--Kutta Time Discretizations. We explore the possibility in avoiding the local characteristic decompositions or even the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining nonoscillatory properties for problems with strong shocks. As a result, the Lax--Wendroff Time Discretization procedure is more cost effective than the Runge--Kutta Time Discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics.
Nikolaos Kazantzis - One of the best experts on this subject based on the ideXlab platform.
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control relevant Discretization of nonlinear systems with Time delay using taylor lie series
Journal of Dynamic Systems Measurement and Control-transactions of The Asme, 2005Co-Authors: Nikolaos Kazantzis, Kilto Chong, Junyoung Park, Alexander G ParlosAbstract:A new Time-Discretization method for the development of a sampled-data representation of a nonlinear continuous-Time input-driven system with Time delay is proposed. It is based on the Taylor-Lie series expansion method and zero-order hold assumption. The mathematical structure of the new Discretization scheme is explored and characterized as useful for establishing concrete connections between numerical and system-theoretic properties. In particular, the effect of the Time-Discretization method on key properties of nonlinear control systems, such as equilibrium properties and asymptotic stability, is examined. The resulting Time-Discretization provides a finite-dimensional representation for nonlinear control systems with Time-delay enabling the application of existing controller design techniques. The performance of the proposed Discretization procedure is evaluated using the case study of a two-degree-of-freedom mechanical system that exhibits nonlinear behavior. Various sampling rates and Time-delay values are considered.
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control relevant Discretization of nonlinear systems with Time delay using taylor lie series
American Control Conference, 2003Co-Authors: Nikolaos Kazantzis, Kilto Chong, Junyoung Park, Alexander G ParlosAbstract:A new Time-Discretization method for the development of a discrete-Time (sampled-data) representation of a nonlinear continuous-Time control system with Time-delay is proposed. It is based on the Taylor-Lie series expansion method and zero-order hold (ZOH) assumption. The mathematical structure of the new Discretization scheme is explored and characterized as useful for establishing concrete connections between numerical and system-theoretic properties. The effect of the Time-Discretization method on key properties of nonlinear control systems, such as equilibrium properties and asymptotic stability, is examined. The resulting Time-Discretization provides a finite-dimensional representation for nonlinear control systems with Time-delay enabling the application of existing controller design techniques. The performance of the proposed Discretization procedure is evaluated using a case study.
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Time Discretization of nonlinear control systems via taylor methods
Computers & Chemical Engineering, 1999Co-Authors: Nikolaos Kazantzis, Costas KravarisAbstract:Abstract A new Discretization method for the calculation of a sampled-data representation of a nonlinear continuous-Time system is proposed. It is based upon the well-known Taylor method and the zero-order hold (ZOH) assumption. The mathematical structure of the new Discretization scheme is analyzed and characterized as being particularly useful in establishing concrete connections between numerical properties and system-theoretic properties. In particular, the effect of the Taylor Discretization procedure on key properties of nonlinear systems, such as equilibrium properties and asymptotic stability, is examined. Within a control context, numerical aspects of Taylor Discretization are also discussed, and ‘hybrid’ Discretization schemes, that result from a combination of the ‘scaling and squaring’ technique with the Taylor method, are also proposed, especially under conditions of very low sampling rates. Practical issues associated with the selection of the method’s parameters to meet CPU Time and accuracy requirements, are examined as well. Finally, the performance of the proposed Discretization procedure is evaluated in a chemical reactor example, that exhibits nonlinear behavior and is subject to various sampling rates.
Sébastien Imperiale - One of the best experts on this subject based on the ideXlab platform.
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Dispersion analysis of improved Time Discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string.
2013Co-Authors: Juliette Chabassier, Sébastien ImperialeAbstract:We study the implicit Time Discretization of Tim- oshenko prestressed beams. This model features two types of waves: flexural and shear waves, that propa- gate with very different velocities. We present a novel implicit Time Discretization adapted to the physical phenomena occuring at the continuous level. Af- ter analyzing the continuous system and the two branches of eigenfrequencies associated with the standing modes, the classical θ-scheme is studied. A dispersion analysis recalls that θ = 1/12 re- duces the numerical dispersion, but yields a severely constrained stability condition for our application. Therefore we propose a new θ-like scheme based on two parameters adapted to each wave velocity, which reduces the numerical dispersion while relaxing this stability condition. Numerical experiments success- fully illustrate the theoretical results on the specific cas of a realistic piano string. This motivates the extension of the proposed approach for more chal- lenging physics.
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Stability and dispersion analysis of improved Time Discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string.
2012Co-Authors: Juliette Chabassier, Sébastien ImperialeAbstract:We study the implicit Time Discretization of piano strings governing equations within the Timoshenko prestressed beam model. Such model features two different waves, namely the flexural and shear waves, that propagate with very different velocities. We present a novel implicit Time Discretization that reduces the numerical dispersion while allowing the use of a large Time step in the numerical computations. After analyzing the continuous system and the two branches of eigenfrequencies associated with the propagating mode{s}, the classical $\theta$-scheme is studied. We present complete {new} proofs of stability using energy-based approaches that provide uniform results with respect to the featured Time step. A dispersion analysis confirm{s} that $\theta=1/12$ reduces the numerical dispersion, but yields a severely constrained stability condition for the application considered. Therefore we propose a new $\theta$-like scheme, which allows to reduce the numerical dispersion while relaxing this stability condition. Stability proofs {are} also provided for this new scheme. Theoretical results {are} illustrated with numerical experiments corresponding to the simulation of a realistic piano string.
Juliette Chabassier - One of the best experts on this subject based on the ideXlab platform.
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Dispersion analysis of improved Time Discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string.
2013Co-Authors: Juliette Chabassier, Sébastien ImperialeAbstract:We study the implicit Time Discretization of Tim- oshenko prestressed beams. This model features two types of waves: flexural and shear waves, that propa- gate with very different velocities. We present a novel implicit Time Discretization adapted to the physical phenomena occuring at the continuous level. Af- ter analyzing the continuous system and the two branches of eigenfrequencies associated with the standing modes, the classical θ-scheme is studied. A dispersion analysis recalls that θ = 1/12 re- duces the numerical dispersion, but yields a severely constrained stability condition for our application. Therefore we propose a new θ-like scheme based on two parameters adapted to each wave velocity, which reduces the numerical dispersion while relaxing this stability condition. Numerical experiments success- fully illustrate the theoretical results on the specific cas of a realistic piano string. This motivates the extension of the proposed approach for more chal- lenging physics.
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Stability and dispersion analysis of improved Time Discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string.
2012Co-Authors: Juliette Chabassier, Sébastien ImperialeAbstract:We study the implicit Time Discretization of piano strings governing equations within the Timoshenko prestressed beam model. Such model features two different waves, namely the flexural and shear waves, that propagate with very different velocities. We present a novel implicit Time Discretization that reduces the numerical dispersion while allowing the use of a large Time step in the numerical computations. After analyzing the continuous system and the two branches of eigenfrequencies associated with the propagating mode{s}, the classical $\theta$-scheme is studied. We present complete {new} proofs of stability using energy-based approaches that provide uniform results with respect to the featured Time step. A dispersion analysis confirm{s} that $\theta=1/12$ reduces the numerical dispersion, but yields a severely constrained stability condition for the application considered. Therefore we propose a new $\theta$-like scheme, which allows to reduce the numerical dispersion while relaxing this stability condition. Stability proofs {are} also provided for this new scheme. Theoretical results {are} illustrated with numerical experiments corresponding to the simulation of a realistic piano string.
Peter Benner - One of the best experts on this subject based on the ideXlab platform.
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fast solution of cahn hilliard variational inequalities using implicit Time Discretization and finite elements
Journal of Computational Physics, 2014Co-Authors: Jessica Bosch, Martin Stoll, Peter BennerAbstract:We consider the efficient solution of the Cahn-Hilliard variational inequality using an implicit Time Discretization, which is formulated as an optimal control problem with pointwise constraints on the control. By applying a semi-smooth Newton method combined with a Moreau-Yosida regularization technique for handling the control constraints we show superlinear convergence in function space. At the heart of this method lies the solution of large and sparse linear systems for which we propose the use of preconditioned Krylov subspace solvers using an effective Schur complement approximation. Numerical results illustrate the competitiveness of this approach.
