This article deals with the numerical solution to some models described by the system of strongly coupled reaction-diffusion equations with the Neumann boundary value conditions. A linearized three-level scheme is derived by the method of reduction of order. The uniquely solvability and second-order convergence in L 2 -norm are proved by the energy method. A numerical example is presented to demonstrate the accuracy and efficiency of the proposed method.
The phase field crystal (PFC) model is a nonlinear evolutionary equation that is of sixth order in space. In the first part of this work, we derive a three level linearized difference scheme, which is then proved to be energy stable, unique solvable and second order convergent in L 2 norm by the energy method combining with the inductive method. In the second part of the work, we analyze the unique solvability and convergence of a two level nonlinear difference scheme, which was developed by Zhang et al. in 2013. Some numerical results with comparisons are provided.
Keywordsphase field crystal model, nonlinear evolutionary equation, finite difference scheme, solvability, convergence
MSC(2010) 65M06, 65M12, 65M15Citation: Cao H Y, Sun Z Z. Two finite difference schemes for the phase field crystal equation.
This article deals with the numerical solution to the magneto-thermo-elasticity model, which is a system of the third order partial differential equations. By introducing a new function, the model is transformed into a system of the second order generalized hyperbolic equations. A priori estimate with the conservation for the problem is established. Then a three-level finite difference scheme is derived. The unique solvability, unconditional stability and second-order convergence in L∞-norm of the difference scheme are proved. One numerical example is presented to demonstrate the accuracy and efficiency of the proposed method.
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