High performance computing for partial differential equations
Introduction
In incompressible fluid flows the PDE solution is constrained by the material conditions , where u is the velocity of the fluid. This constraint can be exactly satisfied, the number of variables of the initial problem can then be reduced, and correct results are obtained [1]. If the constraint is not eliminated but just approximated, the number of degrees of freedom is too high, and spurious modes can appear. When one couples into those unphysical modes, the result can be affected. In Maxwell’s equations, or in magnetohydrodynamics (MHD), the linear spectrum has singular solutions such as infinite degeneracies, accumulation points, or continuous spectra. These singularities are related to internal constraints. These must be correctly approximated [2], otherwise, the degeneracies open into Sturmian spectra, the so-called spectrum pollution appears, and the numerical solution does not stably converge towards the physical one.
In a number of physical phenomena, the solution can show different behaviors on a characteristic surface and orthogonal to it. For instance, a charged particle moves on a magnetic flux surface. When the numerical approach cannot guarantee this, the physics can be strongly affected. For instance, in MHD stability analysis, the magnetic flux surface has to be chosen as an independent variable, and the dependent variables must be chosen on and orthogonal to those surfaces [2]. These are prerequisits to achieve enough precision when predicting stability behaviors of fusion reactor experiments such as ITER [3] or the Wendelstein 7-X Stellerator [4].
High precision can be achieved by a spectral element method [5]. It shows exponential convergence for smooth solutions. It has been shown that it is possible to slightly modify this method to suppress spurious modes and spectral pollution [1]. However, if solutions have to be found that develop δ functions or stepwise solutions, it is advantageous to choose a lowest order, least regular basis function approach [2].
Job run time and energy consumption can be reduced by adapting the computational resource to application characteristics. For this purpose the monitoring system VAMOS has been developed that measures a number of characteristic parameters of the application. A main memory access dominated application can be detected during execution, and the processor frequency can be reduced, thus, reducing energy consumption [6]. This monitoring system is also used to detect wrong complexities and poor parallel implementations.
After execution, the measured quantities are stored in a data base, and reused to perform a complexity analysis needed to predict the CPU and communication times of a new parallel job. Together with information on the status of accessible computers the costs of the submitted application can be evaluated on different available resources. The goal is then to choose the most adequate resource [6]. Such information can also be used to detect resources that are underused and therefore eligible for decommissioning.
Section snippets
COOL method
When a physical phenomenon is described by partial differential equations, we can distinguish between three different types of operators.
Improve precision
Increasing precision by a higher degree approximation method is not always a good idea. If the solution reveals a singular behavior, high degree polynomials develop oscillatory Gibb’s phenomena that can be unacceptable. Then, lowest order elements with lowest regularity lead to well represented delta function like eigensolutions in a continuous spectrum, or to non-oscillatory jumps in unstable kink modes [2].
If we have to deal with multiphysics models, the right choice of the mesh can strongly
Adapt computers to applications
The physics-conforming COOL method together with the right choice of dependent and independent variables, leads to correct and precise results, spectral elements increase precision and reduce run time. Altogether the “science per Watt” ratio can be improved. An additional step towards an energy reduction is to adapt the computer to application needs by intervening on hardware components using knowledge collected by the recently developed VAMOS system [6]. This software is part of the ïanos Grid
Conclusions
Numerical experimentations deliver physically relevant results if the numerical approach can well describe the physical model. Three types of PDE operators have been presented. A Type I operator such as the Laplacian can easily be well represented by any sufficiently regular numerical approximation method. In a Type II operator, for which the eigensolutions are constrained by an external material related condition such as the incompressibility constraint for water, spurious modes can appear if
Acknowledgement
We would like to thank Michel Deville for his continuous encouragement to improve numerical methods, to increase application performance, and to use best suited computational resources. We also take the opportunity to wish him all the best for his retirement. We all hope that he will soon climb the Säntis, one of his latest challenges.
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2020, npj Computational Materials