Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations (PDEs). As other types of fast sweeping schemes, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order. The resulting iterative schemes have a fast convergence rate to steady-state solutions. Moreover, an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system. Hence, they are robust and flexible, and have been combined with high-order accurate weighted essentially non-oscillatory (WENO) schemes to solve various hyperbolic PDEs in the literature. For multidimensional nonlinear problems, high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs. In this technical note, we apply sparse-grid techniques, an effective approximation tool for multidimensional problems, to fixed-point fast sweeping WENO methods for reducing their computational costs. Here, we focus on fixed-point fast sweeping WENO schemes with third-order accuracy (Zhang et al. 2006 [
In this work, we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion, the instantaneous nonlinear cubic Kerr response, and the nonlinear delayed Raman molecular vibrational response. Unlike the first-order PDE-ODE governing equations considered previously in Bokil et al. (J Comput Phys 350: 420–452, 2017) and Lyu et al. (J Sci Comput 89: 1–42, 2021), a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part (i.e., the auxiliary differential equations) modeling the linear and nonlinear dispersion in the material. The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization. A nodal discontinuous Galerkin (DG) method is further applied in space for efficiently handling nonlinear terms at the algebraic level, while preserving the energy stability and achieving high-order accuracy. Indeed with as the number of the components of the electric field, only a nonlinear algebraic system needs to be solved at each interpolation node, and more importantly, all these small nonlinear systems are completely decoupled over one time step, rendering very high parallel efficiency. We evaluate the proposed schemes by comparing them with the methods in Bokil et al. (2017) and Lyu et al. (2021) (implemented in nodal form) regarding the accuracy, computational efficiency, and energy stability, by a parallel scalability study, and also through the simulations of the soliton-like wave propagation in one dimension, as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric (TE) mode of the equations.
The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems. Here, we develop a new analytical approach based on the bifurcation analysis to explore the mathematical solution structure of phase field models. Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions. To elucidate the idea, we apply this analytical approach to three representative phase field equations: the Allen-Cahn equation, the Cahn-Hilliard equation, and the Allen-Cahn-Ohta-Kawasaki system. The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.