Publication

High performance computing for partial differential equations

Abstract

When representing realistic physical phenomena by partial differential equations (PDE), it is crucial to approximate the underlying physics correctly, to get precise results, and to efficiently use the computer architecture. Incorrect results can appear in incompressible Navier-Stokes or Stokes problems when the numerical approach couples into spurious modes. In Maxwell or magnetohydrodynamic (MHD) equations the so-called spectrum pollution effect can occur, and the numerical solution does not stably converge to the physical one. Problems coming from a mesh that is not adapted to the underlying physical problem, or from an inadequate choice of the dependent and independent variables can lead to low precision. Efficiency of a code implementation can be improved by well adapting the parallel computer to the application. A new monitoring system enables to detect poor implementations, to find best suited resources to execute the job, and to adapt the processor frequency during. (C) 2010 Elsevier Ltd. All rights reserved.

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Related concepts (35)
Numerical methods for partial differential equations
Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist. Finite difference method In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.
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