In this paper, we construct high-order stabilization terms for semi-implicit deferred correction methods, focusing on convection-diffusion-reaction problems. Optimal choices for stability parameters are investigated and the upper bounds of the timestep to ensure both stability and high-order accuracy are analyzed. Utilizing Fourier analysis, we apply von Neumann stability and Schur theory to new single- and multi-step deferred correction methods. Numerical experiments validate the analysis of the stability parameter estimates and timestep constraints for these methods.
A nonconforming virtual element scheme for the Allen-Cahn equation on polygonal meshes is developed. The numerical scheme employs a variable-time-step two-step backward differentiation formula (BDF2) for temporal discretization and a lowest-order nonconforming virtual element method in space. With the help of discrete we obtain the error estimates of temporal semi-discrete system under a mild time-step ratio 0 < r(k) < r(max) approximate to 4.8645. By using temporal-spatial error splitting approach, we establish the L-infinity-norm bound of discrete solution, and the unconditional optimal L-2-error estimates are carried out without CFL condition. Numerical experiments validate the theoretical findings across diverse polygonal meshes.
An energy-optimized exponential scalar auxiliary variable (EOP-ESAV) method for the Ginzburg-Landau equation (GLE) with a relaxation step in exponential scalar auxiliary variable (ESAV) schemes is proposed. The EOP-ESAV method is an explicit linear approach that allows for high-order, computationally efficient schemes. Furthermore, this approach retains the convergence order of the ESAV schemes while achieving smaller numerical errors. Most notably, we rigorously prove that the proposed method maintains a modified energy dissipation law that optimally approximates the original energy within the relaxation framework. Extensive numerical experiments are conducted to demonstrate notable improvements of the EOP-ESAV method in accuracy and numerical stability. The EOP-ESAV schemes are also applied on the simulation of vortex dynamics in GLEs with various setups, studying vortex dipole interactions and lattice motion patterns.
As a machine learning method for solving partial differential equations, the random feature method enjoys both the flexibility of neural networks and the fidelity with spectral convergence. However, the partition-of-unity method used in existing works of the random feature method refrains the method from straightforwardly solving partial differential equations of order greater than two, and extra equations for continuity conditions or for order reduction are required. In this paper, we propose the analytic the random feature method to tackle this issue, by applying analytic unit partition-ofunity functions in the random feature method to achieve arbitrary smoothness. Namely, we study a group of partition-of-unity functions constructed using analytic sigmoidal functions, which have exponential decay rate, and can be truncated to the same window like by the existing approach. Numerical tests validate the proposal in solving high-order partial differential equations, and that the proposed analytic unit partitionof-unity functions can replace the default psi b function used by existing works, both with improved precision when the same number of unknowns are solved.
The optimization algorithms are crucial in training physics-informed neural networks (PINNs), as unsuitable methods may lead to poor solutions. Compared to the common gradient descent (GD) algorithm, implicit gradient descent (IGD) outperforms it in handling certain multi-scale problems. In this paper, we provide convergence analysis for the IGD in training over-parameterized two-layer PINNs. We first derive the training dynamics of IGD in training two-layer PINNs. Then, over-parameterization allows us to prove that the randomly initialized IGD converges to a globally optimal solution at a linear convergence rate. Moreover, due to the distinct training dynamics of IGD compared to GD, the learning rate can be selected independently of the sample size and the least eigenvalue of the Gram matrix. Additionally, the novel approach used in our convergence analysis imposes a milder requirement on the network width. Finally, empirical results validate our theoretical findings.
