An efficient combination method of Laplace transform and mixed multiscale finite-element method for coupling partial differential equations of flow in a dual-permeability system is present. First, the time terms of parabolic equation with unknown pressure term are removed by the Laplace transform. Then the transformed equations are solved by mixed FEMs which can provide the numerical approximation formulas for pressure and velocity at the same time. With some assumptions, the multiscale basis functions are constructed by utilizing the effects of fine-scale heterogeneities through basis functions formulation computed from local flow problems. Without time step in discrete process, the present method is efficient when solving spatial discrete problems. At last, the associated pressure transform is inverted by the method of numerical inversion of the Laplace transform.
In this paper, a new numerical method is developed for solving one-dimensional second-order hyperbolic equations.By using a new unconditionally stable two level difference scheme based on the quartic spline interpolation method in space direction and generalized trapezoidal formula in time direction, the hyperbolic equations are solved. Stability analysis of the scheme is carried out. The accuracy of the scheme is second-order in time direction and fourth-order in space direction. It has been shown that by suitably choosing parameter, a high accuracy scheme of third-order accurate in time direction can be derived from the method. Numerical results comparison demonstrate the superiority of the new scheme.
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