11 results
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2. THE REDUCED ORDER NS-α MODEL FOR INCOMPRESSIBLE FLOW: THEORY, NUMERICAL ANALYSIS AND BENCHMARK TESTING.
- Author
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CUFF, VICTORIA M., DUNCA, ARGUS A., MANICA, CAROLINA C., and REBHOLZ, LEO G.
- Subjects
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INCOMPRESSIBLE flow , *NUMERICAL analysis , *BENCHMARK testing (Engineering) , *MESH analysis (Electric circuits) , *ALGORITHMS - Abstract
This paper introduces a new, reduced-order NS-α (rNS-α) model for the purpose of efficient, stable, and accurate simulations of incompressible flow problems on coarse meshes. We motivate the new model by discussing the difficulties in efficient and stable algorithm construction for the usual NS-α model, and then derive rNS-α by using deconvolution as an approximation to the filter inverse, which reduces the fourth order NS-α formulation to a second order model. After proving the new model is well-posed, we propose a C0 finite element spatial discretization together with an IMEX BDF2 timestepping to create a linearized algorithm that decouples the conservation of mass and momentum equations from the filtering. We rigorously prove the algorithm is well-posed, and provided a very mild timestep restriction, is also stable and converges optimally to the model solution. Finally, we give results of several benchmark computations that confirm the theory and show the proposed model/scheme is effective at efficiently finding accurate coarse mesh solutions to flow problems. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
3. A localized orthogonal decomposition method for semi-linear elliptic problems.
- Author
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Henning, Patrick, Målqvist, Axel, and Peterseim, Daniel
- Subjects
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ORTHOGONAL decompositions , *GALERKIN methods , *OSCILLATIONS , *ALGORITHMS , *NUMERICAL analysis - Abstract
In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H
log (H) where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh size even for rough coefficients. To solve the corresponding system of algebraic equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space. [ABSTRACT FROM AUTHOR] - Published
- 2014
- Full Text
- View/download PDF
4. Parallel Schwarz Waveform Relaxation Algorithm for an N-dimensional semilinear heat equation.
- Author
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Tran, Minh-Binh
- Subjects
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HEAT equation , *WAVE analysis , *ALGORITHMS , *STOCHASTIC convergence , *MATHEMATICAL models , *NUMERICAL analysis - Abstract
We present in this paper a proof of well-posedness and convergence for the parallel Schwarz Waveform Relaxation Algorithm adapted to an N-dimensional semilinear heat equation. Since the equation we study is an evolution one, each subproblem at each step has its own local existence time, we then determine a common existence time for every problem in any subdomain at any step. We also introduce a new technique: Exponential Decay Error Estimates, to prove the convergence of the Schwarz Methods, with multisubdomains, and then apply it to our problem. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
5. The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations.
- Author
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Auroux, Didier and Nodet, Maëlle
- Subjects
- *
INVERSE problems , *NUMERICAL analysis , *LINEAR equations , *ALGORITHMS , *VISCOUS flow , *STOCHASTIC convergence , *EXPONENTIAL functions - Abstract
In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers’ equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We show that for non viscous equations (both linear transport and Burgers), the convergence of the algorithm holds under observability conditions. Convergence can also be proven for viscous linear transport equations under some strong hypothesis, but not for viscous Burgers’ equation. Moreover, the convergence rate is always exponential in time. We also notice that the forward and backward system of equations is well posed when no nudging term is considered. [ABSTRACT FROM PUBLISHER]
- Published
- 2012
- Full Text
- View/download PDF
6. Numerical algorithms for backward stochastic differential equations with 1-d brownian motion: Convergence and simulations***.
- Author
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Peng, Shige and Xu, Mingyu
- Subjects
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NUMERICAL analysis , *ALGORITHMS , *STOCHASTIC differential equations , *WIENER processes , *STOCHASTIC convergence , *SIMULATION methods & models - Abstract
In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
7. Total overlapping Schwarz' preconditioners for elliptic problems.
- Author
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Ben Belgacem, Faker, Gmati, Nabil, and Jelassi, Faten
- Subjects
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SCHWARZ function , *ELLIPTIC functions , *ITERATIVE methods (Mathematics) , *ALGORITHMS , *ABSORPTION , *BOUNDARY value problems , *NUMERICAL analysis - Abstract
A variant of the Total Overlapping Schwarz (TOS) method has been introduced in [Ben Belgacem et al., C. R. Acad. Sci., Sér. 1 Math.336(2003) 277–282] as an iterative algorithm to approximate the absorbing boundary condition, in unbounded domains. That same method turns to be an efficient tool to make numerical zooms in regions of a particular interest. The TOS method enjoys, then, the ability to compute small structures one wants to capture and the reliability to obtain the behavior of the solution at infinity, when handling exterior problems. The main aim of the paper is to use this modified Schwarz procedure as a preconditioner to Krylov subspaces methods so to accelerate the calculations. A detailed study concludes to a super-linear convergence of GMRES and enables us to state accurate estimates on the convergence speed. Afterward, some implementation hints are discussed. Analytical and numerical examples are also provided and commented that demonstrate the reliability of the TOS-preconditioner. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
8. A discrete contact model for crowd motion.
- Author
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Maury, Bertrand and Venel, Juliette
- Subjects
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COLLECTIVE behavior , *ADMISSIBLE sets , *MATHEMATICAL analysis , *NP-complete problems , *PREDICTION models , *ALGORITHMS , *NUMERICAL analysis , *SIMULATION methods & models - Abstract
The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: we first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people. The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e.velocities which do not violate the non-overlapping constraint). We describe here the underlying mathematical framework, and we explain how recent results by J.F. Edmond and L. Thibault on the sweeping process by uniformly prox-regular sets can be adapted to handle this situation in terms of well-posedness. We propose a numerical scheme for this contact dynamics model, based on a prediction-correction algorithm. Numerical illustrations are finally presented and discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
9. Numerical study of the systematic error in Monte Carlo schemes for semiconductors.
- Author
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Muscato, Orazio, Wagner, Wolfgang, and Di Stefano, Vincenza
- Subjects
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ERROR analysis in mathematics , *SEMICONDUCTORS , *MONTE Carlo method , *STOCHASTIC convergence , *NUMERICAL analysis , *ELECTRIC fields , *ALGORITHMS , *RUNGE-Kutta formulas - Abstract
The paper studies the convergence behavior of Monte Carlo schemes for semiconductors. A detailed analysis of the systematic error with respect to numerical parameters is performed. Different sources of systematic error are pointed out and illustrated in a spatially one-dimensional test case. The error with respect to the number of simulation particles occurs during the calculation of the internal electric field. The time step error, which is related to the splitting of transport and electric field calculations, vanishes sufficiently fast. The error due to the approximation of the trajectories of particles depends on the ODE solver used in the algorithm. It is negligible compared to the other sources of time step error, when a second order Runge-Kutta solver is used. The error related to the approximate scattering mechanism is the most significant source of error with respect to the time step. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
10. Elliptic equations of higher stochastic order.
- Author
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Lototsky, Sergey V., Rozovskii, Boris L., and Wan, Xiaoliang
- Subjects
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STOCHASTIC orders , *EQUATIONS , *NUMERICAL analysis , *NONLINEAR functional analysis , *CALCULUS , *ALGORITHMS , *STOCHASTIC convergence - Abstract
This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Itô-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of the Wiener Chaos (Cameron-Martin) expansions. The existence and uniqueness of the solutions are established under rather weak assumptions, the main of which requires only that the expectation of the highest order (differential) operator is a non-degenerate elliptic operator. The deterministic coefficients of the Wiener Chaos expansion of the solution solve a lower-triangular system of linear elliptic equations (the propagator). This structure of the propagator insures linear complexity of the related numerical algorithms. Using the lower triangular structure and linearity of the propagator, the rate of convergence is derived for a spectral/hpfinite element approximation. The results of related numerical experiments are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
11. A Domain Decomposition Algorithm for Contact Problems: Analysis and Implementation.
- Author
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Haslinger, J., Kučera, R., and Sassi, T.
- Subjects
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CONTACT mechanics , *APPLIED mechanics , *ALGORITHMS , *ITERATIVE methods (Mathematics) , *NUMERICAL analysis - Abstract
The paper deals with an iterative method for numerical solving frictionless contact problems for two elastic bodies. Each iterative step consists of a Dirichlet problem for the one body, a contact problem for the other one and two Neumann problems to coordinate contact stresses. Convergence is proved by the Banach fixed point theorem in both continuous and discrete case. Numerical experiments indicate scalability of the algorithm for some choices of the relaxation parameter. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
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