An Object Oriented Version of the Kinematic Laplacian Equation Method

Abstract

The kinematic Laplacian equation (KLE) method solves the Navier–Stokes equations by means of its vorticity-velocity formulation. This method has been used to calculate the time dependent flow around moving bodies and other fluid dynamic problems with great success.
The KLE computes the time evolution of the vorticity as an ordinary differential equation (ODE) in each node of the discretized space. The input data for the vorticity transport equation at each time step is provided by a modified version of the Poisson linear partial differential equation for the velocity, called KLE Equation.
This paper presents an object oriented implementation based on a general purpose and high performance framework for solving partial differential equations by the finite and spectral element methods. The framework can interact with different high-performance linear algebra libraries, either for dense or sparse matrices.
Different matrix assembly and boundary condition imposition methods were tested as well as two different solvers in order to find the code version with the best performance. The method was validated against a problem with known analytical solution. Scalability tests were performed to study the behavior of the method as the complexity of the problem increases. Results showing the benefits obtained with this implementation are presented.