We develop efficient and accurate sum-of-exponential (SOE) approximations for the Gaussian using rational approximation of the exponential function on the negative real axis. Six digit accuracy can be obtained with eight terms and ten digit accuracy can be obtained with twelve terms. This representation is of potential interest in approximation theory but we focus here on its use in accelerating the fast Gauss transform (FGT) in one and two dimensions. The one-dimensional scheme is particularly straightforward and easy to implement, requiring only twenty-four lines of MATLAB code. The two-dimensional version requires some care with data structures, but is significantly more efficient than existing FGTs. Following a detailed presentation of the theoretical foundations, we demonstrate the performance of the fast transforms with several numerical experiments.

In this paper we present a coupled Finite Element Method – Boundary Element Method (FEM-BEM) approach for the solution of the free-boundary axi-symmetric plasma equilibrium problem. The proposed method, obtained from an improvement of the Hagenow-Lackner coupling method, allows to efficiently model the equilibrium problem in unbounded domains by discretizing only the plasma region; the external conductors can be modelled either as 2D or 3D models, according to the problem of interest. The paper explores different iterative methods for the solution of the nonlinear Grad-Shafranov equation, such as Picard, Newton-Raphson and Newton-Krylov, in order to provide a robust and reliable tool, able to handle large-scale problems (e.g. high resolution equilibria). This method has been implemented in the FRIDA code (FRee-boundary Integro-Differential Axisimmetric – https://github. om/matteobonotto/ FRIDA), together with a suitable Adaptive Integration Technique (AIT) for the computation of the source term. FRIDA has been successfully tested and validated against experimental data from RFX-mod device, and numerical equilibria of an ITER-like device.

In this paper we propose and analyze a second order accurate numerical scheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme, which ensures the positivity-preserving property, i.e., the numerical value of the phase variable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special form of the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearized stability analysis. A few numerical results, including both the constant-mobility and solution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.

In this paper, we propose a deep spatio-temporal forecasting model (DeepSTF) for multi-site weather prediction post-processing by using both temporal and spatial information. In our proposed framework, the spatio-temporal information is modeled by a CNN (convolutional neural network) module and an encoder-decoder structure with the attention mechanism. The novelty of our work lies in that our model takes full account of temporal and spatial characteristics and obtain forecasts of multiple meteorological stations simultaneously by using the same framework. We apply the DeepSTF model to short-term weather prediction at 226 meteorological stations in Beijing. It significantly improves the short-term forecasts compared to other widely-used benchmark models including the Model Output Statistics method. In order to evaluate the uncertainty of the model parameters, we estimate the confidence intervals by bootstrapping. The results show that the prediction accuracy of the DeepSTF model has strong stability. Finally, we evaluate the impact of seasonal changes and topographical differences on the accuracy of the model predictions. The results indicate that our proposed model has high prediction accuracy.

In this paper, we combine the nonlinear HWENO reconstruction in [43] and the fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the static Hamilton-Jacobi equations in a novel HWENO framework recently developed in [22]. The proposed HWENO frameworks enjoys several advantages. First, compared with the traditional HWENO framework, the proposed methods do not need to introduce additional auxiliary equations to update the derivatives of the unknown function $\phi$. They are now computed from the current value of $\phi$ and the previous spatial derivatives of $\phi$. This approach saves the computational storage and CPU time, which greatly improves the computational efficiency of the traditional HWENO scheme. In addition, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping method is used to update the numerical approximation. It is an explicit method and does not involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves some known issues, including that the iterative number is very sensitive to the parameter $ε$ used in the nonlinear weights, as observed in previous studies. Finally, to further reduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate the good performance of the proposed fixed-point fast sweeping HWENO methods.

Prediction and control of chemical mixing are vital for many scientific areas such as subsurface reactive transport, climate modeling, combustion, epidemiology, and pharmacology. Due to the complex nature of mixing in heterogeneous and anisotropic media, the mathematical models related to this phenomenon are not analytically tractable. Numerical simulations often provide a viable route to predict chemical mixing accurately. However, contemporary modeling approaches for mixing cannot utilize available spatial-temporal data to improve the accuracy of the future prediction and can be compute-intensive, especially when the spatial domain is large and for long-term temporal predictions. To address this knowledge gap, we will present in this paper a deep learning (DL) modeling framework applied to predict the progress of chemical mixing under fast bimolecular reactions. This framework uses convolutional neural networks (CNN) for capturing spatial patterns and long short-term memory (LSTM) networks for forecasting temporal variations in mixing. By careful design of the framework—placement of non-negative constraint on the weights of the CNN and the selection of activation function, the framework ensures non-negativity of the chemical species at all spatial points and for all times. Our DL-based framework is fast, accurate, and requires minimal data for training. The time needed to obtain a forecast using the model is a fraction ($≈ \mathcal{O}(10^{−6}))$ of the time needed to obtain the result using a high-fidelity simulation. To achieve an error of 10% (measured using the infinity norm) for capturing local-scale mixing features such as interfacial mixing, only 24% to 32% of the sequence data for model training is required. To achieve the same level of accuracy for capturing global-scale mixing features, the sequence data required for model training is 64% to 70% of the total spatial-temporal data. Hence, the proposed approach—a fast and accurate way to forecast long-time spatial-temporal mixing patterns in heterogeneous and anisotropic media—will be a valuable tool for modeling reactive-transport in a wide range of applications.

We propose a mass-conservative and monotonicity-preserving characteristic finite element method for solving three-dimensional transport and incompressible Navier-Stokes equations on unstructured grids. The main idea in the proposed algorithm consists of combining a mass-conservative and monotonicity-preserving modified method of characteristics for the time integration with a mixed finite element method for the space discretization. This class of computational solvers benefits from the geometrical flexibility of the finite elements and the strong stability of the modified method of characteristics to accurately solve convection-dominated flows using time steps larger than its Eulerian counterparts. In the current study, we implement three-dimensional limiters to convert the proposed solver to a fully mass-conservative and essentially monotonicity-preserving method in addition of a low computational cost. The key idea lies on using quadratic and linear basis functions of the mesh element where the departure point is localized in the interpolation procedures. The proposed method is applied to well-established problems for transport and incompressible Navier-Stokes equations in three space dimensions. The numerical results illustrate the performance of the proposed solver and support its ability to yield accurate and efficient numerical solutions for three-dimensional convection-dominated flow problems on unstructured tetrahedral meshes.

We propose a computational model to study the growth and spread of bacterial biofilms on interfaces, as well as the action of antibiotics on them. Bacterial membranes are represented by boundaries immersed in a fluid matrix and subject to interaction forces. Growth, division and death of bacterial cells follow dynamic energy budget rules, in response to variations in environmental concentrations of nutrients, toxicants and substances released by the cells. In this way, we create, destroy and enlarge boundaries, either spherical or rod-like. Appropriate forces represent details of the interaction between cells, and the interaction with the environment. We can investigate geometrical arrangements and the formation of porous structures. Numerical simulations illustrate the evolution of top views and diametral slices of small biofilm seeds, as well as the action of antibiotics. We show that cocktails of antibiotics targeting active and dormant cells can entirely eradicate a biofilm.

A positivity-preserving, conservative and entropic numerical scheme is presented for the three-temperature grey diffusion radiation hydrodynamics model. More precisely, the dissipation matrices of the colocalized semi-Lagrangian scheme are defined in order to enforce the entropy production on each species (electron or ion) proportionally to its mass as prescribed in [34]. A reformulation of the model is then considered to enable the derivation of a robust convex combination based scheme. This yields the positivity-preserving property at each sub-iteration of the algorithm while the total energy conservation is reached at convergence. Numerous pure hydrodynamics and radiation hydrodynamics test cases are carried out to assess the accuracy of the method. The question of the stability of the scheme is also addressed. It is observed that the present numerical method is particularly robust.