We propose a model-data asymptotic-preserving neural network (MD-APNN) method to solve the nonlinear gray radiative transfer equations (GRTEs). The system is challenging to be simulated with both the traditional numerical schemes and the vanilla physics-informed neural networks (PINNs) due to the multiscale characteristics. Under the framework of PINNs, we employ a micro-macro decomposition technique to construct a new asymptotic-preserving (AP) loss function, which includes the residual of the governing equations in the micro-macro coupled form, the initial and boundary conditions with additional diffusion limit information, the conservation laws, and a few labeled data. A convergence analysis is performed for the proposed method, and a number of numerical examples are presented to illustrate the efficiency of MD-APNNs, and particularly, the importance of the AP property in the neural networks for the diffusion dominating problems. The numerical results indicate that MD-APNNs lead to a better performance than APNNs or pure Data-driven networks in the simulation of the nonlinear non-stationary GRTEs.

In this paper, we propose a DeepONet structure with causality to represent causal linear operators between Banach spaces of time-dependent signals. The theorem of universal approximations to nonlinear operators proposed in [5] is extended to operators with causalities, and the proposed Causality-DeepONet implements the physical causality in its framework. The proposed Causality-DeepONet considers causality (the state of the system at the current time is not affected by that of the future, but only by its current state and past history) and uses a convolution-type weight in its design. To demonstrate its effectiveness in handling the causal response of a physical system, the Causality-DeepONet is applied to learn the operator representing the response of a building due to earthquake ground accelerations. Extensive numerical tests and comparisons with some existing variants of DeepONet are carried out, and the Causality-DeepONet clearly shows its unique capability to learn the retarded dynamic responses of the seismic response operator with good accuracy.

In this study, we investigate the Shallow Water Equations incorporating source terms accounting for Manning friction and a non-flat bottom topology. Our primary focus is on developing and validating numerical schemes that serve a dual purpose: firstly, preserving all steady states within the model, and secondly, maintaining the late-time asymptotic behavior of solutions, which is governed by a diffusion equation and coincides with a long time and stiff friction limit.

Our proposed approach draws inspiration from a penalization technique adopted in [Boscarino et al., SIAM Journal on Scientific Computing, 2014]. By employing an additive implicit-explicit Runge-Kutta method, the scheme can ensure a correct asymptotic behavior for the limiting diffusion equation, without suffering from a parabolic-type time step restriction which often afflicts multiscale problems in the diffusive limit. Numerical experiments are performed to illustrate high order accuracy, asymptotic preserving, and asymptotically accurate properties of the designed schemes.

High-order schemes enable simulating multi-time-scale problems with relatively large time step sizes for fairly accurate solutions. An unified high-order implicit BDF-$q$ $(q = 3,4,5)$ scheme is developed for the Cahn-Hilliard equation with the gradient flow in $H^{−α},α∈(0,1],$ including the classical case and its fractional variants. Introducing the discrete gradient structure, the resulting implicit BDF-$q$ scheme is presented to satisfy the discrete energy dissipation law, which is actually compatible with the one for the classical case as the order of the fractional Laplacian $α$ approaches 1. The $L^2$ norm error estimate for the BDF-$q$ scheme is rigorously proved by developing a discrete Young-type convolution inequality to deal with the nonlinear term along with the fractional Laplacian. Further, the high-order BDF-$q$ scheme is shown to be less time-consuming compared to the variable-step BDF-2 scheme, while the BDF-5 scheme reduces the CPU time in long-time simulation of coarsening dynamics by almost 80%. Numerical examples also demonstrate that high-order schemes are deemed appealing for long-time slow evolution, while variable-step scheme exhibits more flexibility during phase separation at initial state. In light of this, the variable-step BDF-$q$ scheme utilizing the adaptive time-stepping strategy is implemented to capture both the rapid and slow evolutions of the solutions efficiently and accurately even in high dimensions.

The aging of nuclear reactors presents a substantial challenge within the field of nuclear energy. Consequently, there is a critical demand for field reconstruction techniques capable of obtaining comprehensive spatial data about the condition of nuclear reactors, even when provided with limited observer data. It is worth noting that prior research has often neglected to account for the impact of noise and changes in sensor states that can occur during actual production scenarios. In this paper, the so called Noise and Vibration Tolerant ResNet (NVT-ResNet) is proposed to tackle these challenges. By introducing noise and vibrations into the training data, NVT-ResNet is able to learn the tolerance thus exhibits robustness for the field reconstruction. The influence of sensor numbers on the model’s performance is also investigated. Numerical results convincingly demonstrate that even with limited sparse sensors exposed to a noise with magnitude of 5% and vibrations, NVT-ResNet consistently achieves a reconstruction field of relative $L_2$ error within 1% and relative $L_∞$ error of less than 5% in average sense. Additionally, NVT-ResNet exhibits remarkable computational efficiency, with field reconstruction taking only microseconds. This makes it a viable candidate for integration into online monitoring systems, thereby enhancing the safety performance of nuclear reactors.

In this paper, an improved diffuse-interface lattice Boltzmann method for particulate flows is developed. In this method, the difference in viscosity between the fluid and the particle is considered, and a smooth function is introduced to the collision term of the evolution equation. Some simulations are performed to test the method, and the numerical results show that, compared to the previous diffusion-interface lattice Boltzmann method [J. Liu et al., Comput. Fluids 233 (2022) 105240] which only has a first-order accuracy in space, the present one has a second-order convergence rate in space.

In this paper, we investigate two fundamental geometric properties of the Landau-Lifshitz-Gilbert (LLG) equation, namely the preservation of magnetization magnitude and the Lyapunov structure, by using multistep methods. While the majority of current multistep methods for solving the LLG equation are based on two-step discrete schemes, our research specifically focuses on investigating more general multistep methods. Our proposed methods encompass a range of multistep discrete schemes that allow for achieving any desired order of accuracy in the temporal domain. In this highly general framework, we demonstrate that the magnitude of magnetization is preserved within an error of order $(p+2)$ in time when employing a $(p+1)$th-order multistep discrete scheme. Additionally, the Lyapunov structure is preserved with a first-order error of temporal step size. Finally, some numerical experiments are presented to validate the accuracy of the proposed multistep discrete schemes.

Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology. In this paper, we propose two alternating direction implicit (ADI) schemes to solve the 2D Keller-Segel systems directly with minimal computational cost, while preserving positivity, energy dissipation law and mass conservation. One scheme unconditionally preserves positivity, while the other does so conditionally. Both schemes achieve second-order accuracy in space, with the former being first-order accuracy in time and the latter second-order accuracy in time. Besides, the former scheme preserves the energy dissipation law asymptotically. We validate these results through numerical experiments, and also compare the efficiency of our schemes with the standard five-point scheme, demonstrating that our approaches effectively reduce computational costs.

This paper presents a novel 3-D full electromagnetic particle-in-cell (PIC) code called JefiPIC, which uses Jefimenko’s equations as the electromagnetic (EM) field solver through a full-space integration method. Leveraging the power of state-of-the-art graphic processing units (GPUs), we have made the challenging integral task of PIC simulations achievable. Our proposed code offers several advantages by utilizing the integral method. Firstly, it offers a natural solution for modeling non-neutral plasmas without the need for pre-processing such as solving Poisson’s equation. Secondly, it eliminates the requirement for designing elaborate boundary layers to absorb fields and particles. Thirdly, it maintains the stability of the plasma simulation regardless of the time step chosen. Lastly, it does not require strict charge-conservation particle-to-grid apportionment techniques or electric field divergence amendment algorithms, which are commonly used in finite-difference time-domain (FDTD)-based PIC simulations. To validate the accuracy and advantages of our code, we compared the evolutions of particles and fields in different plasma systems simulated by three other codes. Our results demonstrate that the combination of Jefimenko’s equations and the PIC method can produce accurate particle distributions and EM fields in open-boundary plasma systems. Additionally, our code is able to accomplish these computations within an acceptable execution time. This study highlights the effectiveness and efficiency of JefiPIC, showing its potential for advancing plasma simulations.

We demonstrate a formulation of the Azouani-Olson-Titi (AOT) algorithm in the MPAS-Ocean implementation of the primitive equations of the ocean, presenting global ocean simulations with realistic coastlines and bathymetry. We observe an exponentially fast decay in the error before reaching a certain error level, which depends on the terms involved and whether the AOT feedback control term was handled implicitly or explicitly. A wide range of errors was observed for both schemes, with the implicit scheme typically exhibiting lower error levels, depending on the specific physical terms included in the model. Several factors seem to be contributing to this wide range, but the vertical mixing term is demonstrated to be an especially problematic term. This study provides insight into the promises and challenges of adapting the AOT algorithm to the setting of high-resolution, realistic ocean models.