Notwithstanding the superiority of the Leibniz notation for differential calculus, the dot-and-bar notation predominantly used by the Automatic Differentiation community is resolutely Newtonian. In this paper we exten...
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Productivity-oriented programming languages typically emphasize convenience over syntactic rigor. A well-known example is Matlab, which employs a weak type system to allow the user to assign arbitrary types and shapes...
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The exploitation of sparsity forms an important ingredient for the efficient solution of large-scale problems. For this purpose, this paper discusses two algorithms to detect the sparsity pattern of Hessians: An appro...
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Multiscale differential equations arise in the modeling of many important problems in the science and engineering. Numerical solvers for such problems have been extensively studied in the deterministic case. Here, we ...
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ISBN:
(纸本)9783642219429
Multiscale differential equations arise in the modeling of many important problems in the science and engineering. Numerical solvers for such problems have been extensively studied in the deterministic case. Here, we discuss numerical methods for (mean-square stable) stiff stochastic differential equations. Standard explicit methods, as for example the Euler-Maruyama method, face severe stepsize restriction when applied to stiff problems. Fully implicit methods are usually not appropriate for stochastic problems and semi-implicit methods (implicit in the deterministic part) involve the solution of possibly large linear systems at each time-step. In this paper, we present a recent generalization of explicit stabilized methods, known as Chebyshev methods, to stochastic problems. These methods have much better (mean-square) stability properties than standard explicit methods. We discuss the construction of this new class of methods and illustrate their performance on various problems involving stochastic ordinary and partial differential equations.
The BDDC method [2] is one of the most advanced methods of iterative substructuring. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem has the same st...
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In this paper we present a numerical method to approximate the solution of ID parabolic singularly perturbed problems of reaction-diffusion type. The method combines the Crank-Nicolson scheme and the central finite di...
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ISBN:
(数字)9783642196652
ISBN:
(纸本)9783642196645
In this paper we present a numerical method to approximate the solution of ID parabolic singularly perturbed problems of reaction-diffusion type. The method combines the Crank-Nicolson scheme and the central finite difference scheme defined on nonuniform special meshes. We give a new proof of the asymptotic behavior of the semidiscrete problems resulting after the time discretization. Numerical results show in practice almost second order of uniform convergence of the discrete method.
We investigate Dirichlet-Neumann and Robin methods for a quasilinear elliptic transmission problem in which the nonlinearity changes discontinuously across two subdomains. In one space dimension, we obtain convergence...
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ISBN:
(纸本)9783642113031
We investigate Dirichlet-Neumann and Robin methods for a quasilinear elliptic transmission problem in which the nonlinearity changes discontinuously across two subdomains. In one space dimension, we obtain convergence theorems by extending known results from the linear case. They hold both on the continuous and on the discrete level. From the proofs one can infer mesh-independence of the convergence rates for the Dirichlet-Neumann method, but not for the Robin method. In two space dimensions, we consider numerical examples which demonstrate that the theoretical results might be extended to higher dimensions. Moreover, we investigate the asymptotic convergence behaviour for fine mesh sizes quantitatively. We observe a good agreement with many known linear results, which is remarkable in view of the nonlinear character of the problem.
Wall roughness affects flow characteristics practically in all technical applications. In internal flows, the height of rough elements should be much smaller than the thickness of the shear layer (so called distribute...
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