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Creación y validación de estructuras de datos para aproximar EDP con diferencias finitas en mallas adaptativas
Ordóñez Rodríguez, Claudia Patricia
Ordóñez Rodríguez, Claudia Patricia
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Abstract
Diferentes aplicaciones relacionadas con la física pueden ser modeladas matemáticamente por Ecuaciones Diferenciales Parciales (EDP) como la ecuación de Poisson. Este tipo de ecuaciones con condiciones frontera de Dirichlet y Newmann en el caso bidimensional son aproximadas con métodos numéricos como el método de Diferencias Finitas (DF), especialmente en Mallas con Refinamiento Adaptativo (AMR, abreviatura en inglés de Adaptive Mesh Refinement) en regiones específicas por sus posibles restricciones en su solución.
En esta investigación se lleva a cabo un estudio teórico y computacional de la discretización del problema de Poisson en 2D de forma matricial utilizando el método de DF, tanto en mallas uniformes como en AMR. Se emplean estructuras de datos como Tabla de dispersión (HT, abreviatura en inglés de Hash Table) para representar las células de la malla, lo que implica un análisis detallado de las características de estas estructuras, así como las de sus operaciones y su complejidad computacional. Además, se muestra una variación de métodos de interpolación que permiten la comunicación de las células de los diferentes niveles. Entre los resultados obtenidos, se verifica la convergencia y se realiza un estudio de diferentes propiedades de la matriz como número de condicionamiento, patrón de esparcidad, que se refiere al gráfico de puntos de elementos no nulos de las matrices dispersas, y relación de la interpolación aplicada. Adicionalmente, se lleva a cabo una generación de AMR con varios criterios de refinamiento, mostrando un análisis de eficiencia y de colisiones para las HT.
Different applications related to physics can be mathematically modeled by Partial Differential Equations (PDEs) such as the Poisson's equation. This type of equations, with Dirichlet and Newmann boundary conditions in the two-dimensional case, are approximated with numerical methods such as the Finite Difference (FD) method, especially in Adaptive Meshes Refinement (AMR) in specific regions due to their possible restrictions in their solution. This research conducts a theoretical and computational study of the discretization of the 2D Poisson problem in matrix form using the Finite Difference (FD) method, both on uniform meshes and on Adaptive Mesh Refinement (AMR). Data structures such as Hash Tables (HT) are employed to represent the mesh cells, which involves a detailed analysis of the characteristics of these structures, as well as their operations and computational complexity. Furthermore, a variation of interpolation methods is presented, allowing communication among cells at different levels. Among the results obtained, convergence is verified, and a study of different properties of the matrix such as condition number, sparsity pattern, which refers to the plot of non-zero elements in sparse matrices, and the relationship of the applied interpolation is conducted. Additionally, an AMR generation is carried out using various refinement criteria, showing an analysis of efficiency and collisions for the HT.
Different applications related to physics can be mathematically modeled by Partial Differential Equations (PDEs) such as the Poisson's equation. This type of equations, with Dirichlet and Newmann boundary conditions in the two-dimensional case, are approximated with numerical methods such as the Finite Difference (FD) method, especially in Adaptive Meshes Refinement (AMR) in specific regions due to their possible restrictions in their solution. This research conducts a theoretical and computational study of the discretization of the 2D Poisson problem in matrix form using the Finite Difference (FD) method, both on uniform meshes and on Adaptive Mesh Refinement (AMR). Data structures such as Hash Tables (HT) are employed to represent the mesh cells, which involves a detailed analysis of the characteristics of these structures, as well as their operations and computational complexity. Furthermore, a variation of interpolation methods is presented, allowing communication among cells at different levels. Among the results obtained, convergence is verified, and a study of different properties of the matrix such as condition number, sparsity pattern, which refers to the plot of non-zero elements in sparse matrices, and the relationship of the applied interpolation is conducted. Additionally, an AMR generation is carried out using various refinement criteria, showing an analysis of efficiency and collisions for the HT.
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2024-05-10
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Keywords
Hash table, Numerical analysis, Adaptive Mesh Refinement