Finite difference method

Abstract: A full finite difference time domain methodology is developed for electromagnetic wave propagation in a plasma The finite difference grid is consistent with central difference approximation of the curl, divergence and gradient operators that appear in the joint equations of Euler and Maxwell, and the coupling effects between the fluid velocity and the electric field To accomplish the time advancement, the central difference approximation is invoked for the time derivatives and leapfrog concepts are employed The resulting difference equations converge to the exact equations, provided that the developed stability requirement is satisfied Finally, numerical results are provided and compared with the inverse fast Fourier transform results of closed-form, frequency domain solutions for the half space problem; the agreement between solutions is shown to be excellent

Abstract: For physical phenomena governed by the Biot model of porous-elasticity, a reciprocal relation, similar to the Betti's recoprocal theorem in elasticity, is constructed in Laplace transformed space. Integrating the reciprocal relation enables one to formulate boundary integral equations. The fundamental kernels for the integral equations are solved in closed forms for the case of isotropic material. Numerical implementation of two-dimensional problems includes finite element ideas of discretization and polynomial interpolation, and numerical inversion of a Laplace transform. Practical applications of the method are found in consolidation problems in soils which contain compressible as well as incompressible pore fluids. Also, as a numerical experiment, consolidation of partially saturated soil is simulated and interesting phenomena are observed. The currently developed boundary integral equation method (BIEM) for porous-elasticity may be viewed as an efficient and accurate alternative of existing finite element and finite difference methods. For linear consolidation problems, application of BIEM is always preferred to the other numerical methods whenever possible.

Abstract: We present an error analysis for an unconditionally energy stable, fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equa- tion, a modified Cahn-Hilliard equation coupled with the Darcy flow law. The scheme, proposed by S. M. Wise, is based on the idea of convex splitting. In this paper, we rigorously prove first order convergence in time and second or- der convergence in space. Instead of the (discrete)L ∞ (0,T;L 2 )∩L 2 (0,T;H 2 h ) error estimate, which would represent the typical approach, we provide a dis- creteL ∞ (0,T;H 1 )∩L 2 (0,T;H 3 h )e rror estimate for the phase variable, which allows us to treat the nonlinear convection term in a straightforward way. Our convergence is unconditionalin the sense that the time stepsis in no way constrained by the mesh spacingh .T his is accomplished with the help of anL 2 (0,T;H 3 h ) bound of the numerical approximation of the phase variable. To facilitate both the stability and convergence analyses, we establish a finite difference analog of a Gagliardo-Nirenberg type inequality.