This paper presents a transformed L1 (TL1) finite difference method for the time-fractional coupled nonlinear Schrödinger system. Unconditionally optimal $L^2$ error estimates of the fully discrete scheme are obtained. The convergence results indicate that the method has an order of $2$ in the spatial direction and an order of $2 − α$ in the temporal direction. The error estimates hold without any spatial-temporal stepsize restriction. Such convergence results are obtained by applying a novel discrete fractional Grönwall inequality and the corresponding Sobolev embedding theorems. Numerical experiments for both two-dimensional and three-dimensional models are carried out to confirm our theoretical findings.

When DeepONet approximates solution operators of partial differential equations (PDEs) with discontinuous solutions, it poses a foundational approximation lower bound due to its linear reconstruction property. Inspired by the moving mesh method, we propose an R-adaptive DeepONet method, which consists of: (1) the output data representation is transformed from the physical domain to the computational domain using the equidistribution principle; (2) the maps from input parameters to the solution and the coordinate transformation function over the computational domain are learned using DeepONets separately; (3) the solution over the physical domain is obtained via post-processing methods such as the interpolation method. Additionally, we introduce a solution-dependent weighting strategy in the training process to reduce the error. We establish an upper bound for the reconstruction error based on piecewise linear interpolation and show that the introduced R-adaptive DeepONet can reduce this bound. Moreover, for two prototypical PDEs with sharp gradients or discontinuities, we prove that the approximation error decays at a superlinear rate with respect to the trunk basis size, unlike the linear decay observed in vanilla DeepONets. Numerical experiments on several PDEs with discontinuous solutions are conducted to verify the advantages of the R-adaptive DeepONet over available variants of DeepONet.

We construct first- and second-order time semi-discretization numerical schemes for the Cahn-Hilliard-Navier-Stokes model. This discretization scheme is based on the energy form of the scalar auxiliary variable approach for the coupling terms of model and pressure correction in the Navier-Stokes equations, which are fully decoupled. Then, we apply the fully explicit forms and the two scalar auxiliary variables to obtain stable unconditional energy over time. At the same time, we present the error analysis for the first-order scheme and the convergence rate for all relevant variables in different norms. Finally, numerical examples are presented to validate the theoretical analysis.

This work focuses on the efficient numerical simulation of the Cahn-Hilliard phase-field model of diblock copolymers on evolving surfaces. The model integrates Cahn-Hilliard dynamic of diblock copolymers with partial differential equation on evolving surfaces, which has the property of geometric complexity, nonlinearity and mass conservation. In the numerical simulation, the space-time discretization of the proposed model is realized by the evolving surface finite element method. The stabilized semi-implicit approach is included in the framework of the evolving surface finite element method to produce a linear, stable, conserved and high-accurate scheme for long time numerical simulations. The stability analysis of the designed numerical method is established. Through several numerical experiments, the convergence and stability of the numerical method are investigated. In addition, spinodal decomposition is performed to research the mass evolution and dynamics of the Cahn-Hilliard model of diblock copolymers on different evolving surfaces.

In this paper, we propose and analyze a second order accurate (in time) mass lumped mixed finite element numerical scheme for the liquid thin film coarsening model with a singular Leonard-Jones energy potential. The backward differentiation formula (BDF) stencil is applied in the temporal discretization, and a convex-concave decomposition is derived, so that the concave part corresponds to a quadratic energy. In turn, the Leonard-Jones potential term is treated implicitly and the concave part is approximated by a second order Adams-Bashforth explicit extrapolation. An artificial Douglas-Dupont regularization term is added to ensure the energy stability. Furthermore, we provide a theoretical justification that this numerical algorithm has a unique solution, such that the positivity property is always preserved for the phase variable at a point-wise level, so that a singularity is avoided in the scheme. In fact, the singular nature of the Leonard-Jones potential term around the value of 0 and the mass lumped FEM approach play an essential role in the positivity-preserving property in the discrete level. In addition, an optimal rate convergence estimate in the $ℓ^∞(0, T ; H^{−1}_h )∩ ℓ^2 (0, T ; H^1_h )$ norm is presented. Finally, two numerical experiments are carried out to verify the theoretical properties.

In the conforming discontinuous Galerkin method, the standard bilinear form for the conforming finite elements is applied to discontinuous finite elements without adding any inter-element nor penalty form. The $P_k (k ≥ 1)$ discontinuous finite elements and the $P_{k−1}$ weak Galerkin finite elements are adopted to approximate the velocity and the pressure respectively, when solving the Stokes equations on triangular or tetrahedral meshes. The discontinuous finite element solutions are divergence-free and surprisingly H-div functions on the whole domain. The optimal order convergence is achieved for both variables and for all $k ≥ 1.$ The theory is verified by numerical examples.

This paper proposes a discussion of the direct discontinuous Galerkin (DDG) methods coupled with explicit-implicit-null time discretizations (EIN) for solving the nonlinear diffusion equation $u_t = (a(u)u_x)_x.$ The basic idea of the EIN method is to add and subtract two equal constant coefficient terms $a_1u_{xx}$ $(a_1 = a_0 ×{\rm max}_u a(u))$ on the right-hand side of the above equation, and then apply the explicit-implicit time-marching method to the equivalent equation. The EIN method does not require any nonlinear iterative solver while eliminating the severe time-step restrictions typically associated with explicit methods. We present the stability criterion of the EIN-DDG schemes for the simplified equation with periodic boundary conditions via the Fourier method, where the first order and second order EIN-DDG schemes are unconditionally stable when $a_0 ≥ 0.5$ and the third order EIN-DDG scheme is unconditionally stable under the condition $a_0 ≥ 0.54.$ Numerical experiments show the stability and optimal orders of accuracy of our proposed schemes with a relaxed time-step restriction and the appropriate coefficient $a_0$ for both linear and nonlinear equations in one-dimensional and two-dimensional settings. Furthermore, we also show that the computational efficiency of our EIN-DDG schemes and explicit Runge-Kutta DDG (EX-RK-DDG) schemes for steady-state equations with small viscosity coefficients to illustrate the effectiveness of the present methods.

In this paper, we propose a novel algorithm for finding Cheeger cuts via 1-Laplacian of graphs. In [6], Chang introduced the theory of 1-Laplacian of graphs and built the connection between searching for the Cheeger cut of an undirected and unweighted graph and finding the first nonzero eigenvalue of 1-Laplacian, the latter of which is equivalent to solving a constrained non-convex optimization problem. We develop an alternating direction method of multipliers based algorithm to solve the optimization problem. We also prove that the generated sequence is bounded and it thus has a convergent subsequence. To find the goal optimal solution to the problem, we apply the proposed algorithm using different initial guesses and select the cut with the smallest cut value as the desired cut. Experimental results are presented for typical graphs, including Petersen’s graph and Cockroach graphs, and the well-known Zachary karate club graph.

In this paper, a quadratic spline collocation (QSC) method combined with $L1$ time discretization in the framework of alternating direction implicit (ADI) approach, namely ADI-QSC-$L$1 method, is developed to solve the variable-order time-fractional mobile-immobile diffusion equations in multi-dimensional spaces. Discrete $L_2$ norm-based stability and error estimate are carefully discussed, which show that the proposed method is unconditionally stable and convergent with first-order accuracy in time and second-order accuracy in space. Then, based on the exponential-sum-approximation technique for the fast evaluation of the variable-order Caputo fractional derivative, an efficient implementation strategy of the ADI-QSC-$L1$ method, named ADI-QSC-${\rm F}L1$ is presented, which further improves the computational efficiency by reduced memory requirement and computational cost. Finally, numerical examples are provided to support both the theoretical results and efficiency of the developed method.

A new equivalent reformulation of the absolute value equations associated with second-order cone (SOCAVEs) is emphasised, from which two dynamical models based on projection operator for solving SOCAVEs are constructed. Under suitable conditions, it is proved that the equilibrium points of the dynamical systems exist and could be (globally) asymptotically stable. The effectiveness of the proposed methods are illustrated by some numerical simulations.