Conservation laws of advection-diffusion-reaction (ADR) type are ubiquitous in continuum physics. In this paper we outline discretization of these problems and iterative schemes for the resulting linear system. For di...
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ISBN:
(纸本)9783319399294;9783319399270
Conservation laws of advection-diffusion-reaction (ADR) type are ubiquitous in continuum physics. In this paper we outline discretization of these problems and iterative schemes for the resulting linear system. For discretization we use the finite volume method in combination with the complete flux scheme. The numerical flux is the superposition of a homogeneous flux, corresponding to the advection-diffusion operator, and the inhomogeneous flux, taking into account the effect of the source term (ten Thije Boonkkamp and Anthonissen, J Sci Comput 46(1): 47-70, 2011). For a three-dimensional conservation law this results in a 27point coupling for the unknown as well as the source term. Direct solution of the sparse linear systems that arise in 3D ADR problems is not feasible due to fill-in. Iterative solution of such linear systems remains to be the only efficient alternative which requires less memory and shorter time to solution compared to direct solvers. Iterative solvers require a preconditioner to reduce the number of iterations. We present results using several different preconditioning techniques and study their effectiveness.
As data has become easier to collect and precise sensors have become ubiquitous, data mining with large data sets has become an important problem. Because sparse grid data mining scales only linearly in the number of ...
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ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
As data has become easier to collect and precise sensors have become ubiquitous, data mining with large data sets has become an important problem. Because sparse grid data mining scales only linearly in the number of data points, large data mining problems have been successfully addressed with this method. Still, highly efficient algorithms are required to process very large problems within a reasonable amount of time. In this paper, we introduce a new algorithm that can be used to solve regression and classification problems on spatially adaptive sparse grids. Additionally, our approach can be used to efficiently evaluate a spatially adaptive sparse grid function at multiple points in the domain. In contrast to other algorithms for these applications, our algorithm fits well to modern hardware and performs only few unnecessary basis function evaluations. We evaluated our algorithm by comparing it to a highly efficient implementation of a streaming algorithm for sparse grid regression. In our experiments, we observed speedups of up to 7x, being faster in all experiments that we performed.
In this survey article, we will give a short summary of the basic algorithm issues of discontinuous Galerkin methods for time-dependent convection dominated problems. We will then give a few representative examples of...
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ISBN:
(纸本)9783319416403;9783319416380
In this survey article, we will give a short summary of the basic algorithm issues of discontinuous Galerkin methods for time-dependent convection dominated problems. We will then give a few representative examples of recent developments of discontinuous Galerkin methods for such problems, and provide comparisons with several other types of numerical methods commonly used for similar or related problems. For the comparison, we concentrate mainly on the methods presented in the London Mathematical Society EPSRC Durham Symposium on Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations.
A numerical investigation of unsteady, two-dimensional double diffusive convection flow through a lid-driven square enclosure is carried on. The left and bottom walls of the enclosure are either uniformly or non-unifo...
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ISBN:
(纸本)9783319399294;9783319399270
A numerical investigation of unsteady, two-dimensional double diffusive convection flow through a lid-driven square enclosure is carried on. The left and bottom walls of the enclosure are either uniformly or non-uniformly heated and concentrated, while the right vertical wall is maintained at a constant cold temperature. The top wall is insulated and it moves to the right with a constant velocity. The numerical solution of the coupled nonlinear differential equations is based on the use of dual reciprocity boundary element method (DRBEM) in spatial discretization and an unconditionally stable backward implicit finite difference scheme for the time integration. Due to the coupling and the nonlinearity, an iterative process is employed between the equations. The boundary only nature of the DRBEM and the use of the fundamental solution of Laplace equation make the solution process computationally easier and less expensive compared to other domain discretization methods. The study focuses on the effects of uniform and non-uniform heating and concentration of the walls for various values of physical parameters on the double-diffusive convection in terms of streamlines, isotherms and isoconcentration lines.
The today's demands for simulation and optimization tools for water supply networks are permanently increasing. Practical computations of large water supply networks showthat rather small time steps are needed to ...
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ISBN:
(纸本)9783319399294;9783319399270
The today's demands for simulation and optimization tools for water supply networks are permanently increasing. Practical computations of large water supply networks showthat rather small time steps are needed to get sufficiently good approximation results - a typical disadvantage of low order methods. Having this application in mind we use higher order time discretizations to overcome this problem. Such discretizations can be achieved using so-called strong stability preserving Runge-Kutta methods which are especially designed for hyperbolic problems. We aim at approximating entropy solutions and are interested in weak solutions and variational formulations. Therefore our intention is to compare different space discretizations mostly based on variational formulations, and combine them with a second-order two-stage SDIRK method. In this paper, we will report on first numerical results considering scalar hyperbolic conservation laws.
In this paper a reduced-order strategy is applied to solve a multiobjective optimal control problem governed by semilinear parabolic partial differential equations. These problems often arise in practical applications...
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ISBN:
(纸本)9783319399294;9783319399270
In this paper a reduced-order strategy is applied to solve a multiobjective optimal control problem governed by semilinear parabolic partial differential equations. These problems often arise in practical applications, where the quality of the system behaviour has to be measured by more than one criterium. The weighted sum method is exploited for defining scalar-valued nonlinear optimal control problems built by introducing additional optimization parameters. The optimal controls corresponding to specific choices of the optimization parameters are efficiently computed by the reduced-basis method. The accuracy is guaranteed by an a-posteriori error estimate.
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