A new compact finite difference scheme for solving the complex Ginzburg-Landau equation

a b s t r a c t The complex Ginzburg-Landau equation is often encountered in physics and engineering applications, such as nonlinear transmission lines, solitons, and superconductivity. However, it remains a challenge to develop simple, stable and accurate finite difference schemes for solving the equation because of the nonlinear term. Most of the existing schemes are obtained based on the Crank-Nicolson method, which is fully implicit and must be solved iteratively for each time step. In this article, we present a fourth-order accurate iterative scheme, which leads to a tri-diagonal linear system in 1D cases. We prove that the present scheme is unconditionally stable. The scheme is then extended to 2D cases. Numerical errors and convergence rates of the solutions are tested by several examples

Higher-Order Accurate Finite Difference Method for Thermal Analysis in Double-Layered Solid Structures -- Gradient Preserved Method

Layered structures have appeared in many systems such as biological tissues, micro-electronic devices, thin films, fins, reactor walls, thermoelectric power conversion, thermal coating, metal oxide semiconductors, and thermal processing of DNA origami nanostructures. Analyzing heat transfer in layered structures is of crucial importance for the design and operation of devices and the optimization of thermal processing of materials. There are many numerical methods dealing with the layered structures or interface problems. The existing numerical methods such as the immersed interface method and the matched interface boundary method, if using threegrid points across the interface, usually provide only a second-order truncation error, which reduces the accuracy of the overall numerical solution even if the higher-order compact finite difference method is employed at other points. Obtaining a higherorder accurate numerical scheme using three-grid points across the interface so that the overall numerical scheme is stable and convergent with higher-order accuracy is mathematically challenging. The objective of this dissertation is to develop a higher-order accurate finite difference method using three-grid points across the interface. To this end, we first consider three mathematical models, the steady-state heat conduction model, the unsteady-state heat conduction model and the nanoscale heat conduction model. The well-posedness of these three models are proved. After that, compact higher-order finite difference schemes for solving these three models are developed, respectively. In particular, for the interior points, the well-known pad´e scheme (three-point fourthorder compact finite difference scheme) is applied. On the boundary and interface, by preserving the first-order derivative, ux, fourth-order finite difference schemes for the interface conditions, third-order or fourth-order finite difference schemes for the Neumann boundary conditions and the Robin boundary conditions, are developed, respectively. As such, the overall schemes are at least third-order accurate. The stability and convergence of the scheme for the steady-state case with Dirichlet boundary are proven. Finally, four different examples are given to test the obtained numerical schemes. Results showed that the convergence rate is close to 4.0, which coincides with the theoretical analysis. Further research will focus on the analysis of the stability and convergence of the schemes for the unsteady-state heat conduction case and the nanoscale heat conduction case, and the extension of our schemes to multidimensional cases