Método de elemento finito discontinuo Padé-LDG para la ecuación de Schrödinger
DOI:
https://doi.org/10.31349/RevMexFis.72.020701Keywords:
Schrödinger equation, discontinuous finite element methods, LDG method, Padé approximations, conservative methodsAbstract
Se describe un metodo de alta precisión numérica para la aproximación de la solución de la ecuación de Schrödinger con potenciales independientes del tiempo. El esquema combina el método de elemento finito discontinuo ”Local Discontinuous Galerkin” para la discretización espacial y aproximaciones racionales [m/m]-Pade para el avance en tiempo. Se analiza la conservación de los análogos discretos de la probabilidad y energía. Los experimentos numéricos validan la conservación de los invariantes y además muestran que para problemas suficientemente regulares el método converge con orden τ 2m + h p+1, donde h y τ son los parámetros de discretización en espacio y tiempo, respectivamente; y p es el grado de los polinomios en la aproximación espacial.
A highly accurate numerical method is described for the approximation of the solution of the Schrödinger equation with time-independent potentials. The scheme combines the Local Discontinuous Galerkin finite element method for spatial discretization and rational [m/m]-Padé approximations as time advancing scheme. The conservation of the discrete analogue of the probability and energy is analyzed. Numerical experiments validate the conservation of the invariants and also show that for sufficiently regular problems the method converges with order τ 2m + hp+1 , where h and τ are the discretization parameters in space and time, respectively; and p is the polynomial degree of the spatial approximation.
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