Convergencia del método mimético para la ecuación de difusión no estática
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Date
2023
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Universidad Nacional de Trujillo
Abstract
En el presente trabajo titulado “CONVERGENCIA DEL MÉTODO MIMÉTICO PARA LA ECUACIÓN DE DIFUSIÓN NO ESTÁTICA”, se analiza la convergencia del método mimético para la Ecuación de Difusión No Estática del calor Unidimensional, en tal sentido se realiza la discretización mimética de los operadores Gradiente y Divergencia, en una malla escalonada (punto distribuida); así como de la variable temporal usando el esquema de Crank – Nickolson.
Los operadores que se discretizan en el Método Mimético satisfacen propiedades de operadores diferenciales continuos, así como el Teorema de Green y el Teorema de la Divergencia; por esto se garantiza el aspecto conservativo del método.
Se realiza la implementación computacional de los operadores miméticos unidimensionales y del esquema mimético unidimensional en el caso estático y no estático. Además, se analiza la convergencia del modelo de Diferencias Finitas Miméticas.
Los resultados obtenidos muestran la ventaja del método mimético frente al esquema de Diferencias Finitas, ya que presenta una tasa de convergencia cuadrática, error de truncamiento de segundo orden y mejor aproximación numérica a la solución exacta.
In the present work entitled “CONVERGENCE MIMETIC METHOD FOR THE NOT STATIC DIFFUSION EQUATION “, the convergence of the mimetic method for the Non – Static Diffusion Equation of One-dimensional heat is analyzed, in this sense, the mimetic discretization of the Gradient and Divergence operators, in a staggered mesh (distributed point); as well as the temporary variable using the Crank – Nickolson scheme. The operators that are discretized in the Mimetic Method satisfy properties of continuous differential operators, as well as Green’s Theorem and the Divergence Theorem; this is why the conservative aspect of the method is guaranteed. The computational implementation of the one-dimensional mimetic operators and the one-dimensional mimetic schema in the static and non-static case is performed. In addition, the convergence of the Mimetic Finite Difference model is analyzed. The results obtained show the advantage of the Mimetic Method over the Finite Difference Scheme, since it presents a quadratic convergence rate, second order truncation error and a better numerical approximation to the exact solution.
In the present work entitled “CONVERGENCE MIMETIC METHOD FOR THE NOT STATIC DIFFUSION EQUATION “, the convergence of the mimetic method for the Non – Static Diffusion Equation of One-dimensional heat is analyzed, in this sense, the mimetic discretization of the Gradient and Divergence operators, in a staggered mesh (distributed point); as well as the temporary variable using the Crank – Nickolson scheme. The operators that are discretized in the Mimetic Method satisfy properties of continuous differential operators, as well as Green’s Theorem and the Divergence Theorem; this is why the conservative aspect of the method is guaranteed. The computational implementation of the one-dimensional mimetic operators and the one-dimensional mimetic schema in the static and non-static case is performed. In addition, the convergence of the Mimetic Finite Difference model is analyzed. The results obtained show the advantage of the Mimetic Method over the Finite Difference Scheme, since it presents a quadratic convergence rate, second order truncation error and a better numerical approximation to the exact solution.
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Keywords
Diferencias finitas, Discretización mimética, Ecuación de difusión no estática