Simulations in Science and Engineering for University Education

Abstract

In this paper, the strategies and tests that guarantee a reliable solution of partial differential equations (PDE) solved with the finite difference method are proposed. The elements to be considered are presented so that the solution of equations in partial derivatives is the closest to reality. The experience of the proponents and previous work play an important role in validating simulations in science and engineering problems. A series of tests are proposed that must be carried out to validate and make the simulation results reliable, due to the fact that the selection of the finite difference scheme, the initial data, the boundary conditions and the restrictions in the time step, the stability and convergence of the solution cannot be guaranteed a priori. There are situations in which the solution is stable but converges to values ??that have no physical meaning. To illustrate the strategies and tests inherent in a simulation, such that the solutions can be trusted, the 3D wave equation (three spatial dimensions and one temporal dimension) is solved numerically, by implementing a code in FORTRAN 90, implementing a scheme finite difference to second order approximation. The stability and convergence of the solution are studied according to the schemes proposed in this research, in such a way that the results are reliable, and guarantee the possible practical implementation of the solutions. Finally, the evolution of the scalar field for different times is shown, validated with the principle of conservation of energy. The strategies implemented in this work to guarantee the reliability of the simulations are also valid when solving EDP problems with neural networks, genetic algorithms and other intelligent computing techniques.