A Nonlinear Differential Equation Solver With Potential Application to Pelton Turbines
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The focus in this thesis is the development and implementation of a new method for solving nonlinear differential equations on a grid.
The method’s novelty lies in the way it represents continuously distributed variables by discrete information stored in a grid. The grid contains information about both the values and the values of the derivatives of the unknown functions at the grid points in the computational domain. With this method the derivatives are thus explicitly deﬁned at each grid point rather than, as in conventional numerical schemes, implicitly given by the function values at the surrounding grid points.
By using piecewise polynomial interpolation, functions can be represented with an arbitrary order of continuity over the entire computational domain.
A mathematical framework is deﬁned and the details of the polynomial interpolation is discussed, leading to the deﬁnition of particular sets of basis function which have especially favorable numerical properties for use with the current method.
It is shown how this method is used to formulate sets of differential equations as algebraic equations. With special focus on the Navier-Stokes equations.
A solution algorithm is developed for parallel computation on graphics processor units using C++ and OpenCL. Tests are performed on low cost hardware, and the imple- mentation is developed to meet constraints in memory capacity and processing speed. The algorithm is a residual-minimizing type and is based on the ﬁnite element method. Test cases in 1D, 2D and 3D are numerically solved using the developed code. Two conference presentations are based on the method developed in this thesis, showing ap- plication to simulation of a rising bubble under gravity in three dimensions and a ﬂuid ﬂow in a Pelton turbine cup, also in three dimensions.