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We propose an easy-to-implement iterative method for resolving the implicit (or semi-implicit) schemes arising in solving reaction-diffusion (RD) type equations. We formulate the nonlinear time implicit scheme as a min-max saddle point problem and then apply the primal-dual hybrid gradient (PDHG) method. Suitable precondition matrices are applied to the PDHG method to accelerate the convergence of algorithms under different circumstances. Furthermore, our method is applicable to various discrete numerical schemes with high flexibility. From various numerical examples tested in this paper, the proposed method converges properly and can efficiently produce numerical solutions with sufficient accuracy. [ABSTRACT FROM AUTHOR]

This paper deals with the implementation of high order semi-Lagrangian particle methods to handle high dimensional Vlasov–Poisson systems. It is based on recent developments in the numerical analysis of particle methods and the paper focuses on specific algorithmic features to handle large dimensions. The methods are tested with uniform particle distributions in particular against a recent multi-resolution wavelet based method on a 4D plasma instability case and a 6D gravitational case. Conservation properties, accuracy and computational costs are monitored. The excellent accuracy/cost trade-off shown by the method opens new perspective for accurate simulations of high dimensional kinetic equations by particle methods. [ABSTRACT FROM AUTHOR]

This paper introduces two families of A-stable algorithms for the integration of y ′ = F ( y , t ) y : the extended predictor–corrector (EPC) and the exponential–linear (EL) methods. The structure of the algorithm families are described, and the method of derivation of the coefficients presented. The new algorithms are then tested on a simple deterministic problem and a Monte Carlo isotopic evolution problem. The EPC family is shown to be only second order for systems of ODEs. However, the EPC-RK45 algorithm had the highest accuracy on the Monte Carlo test, requiring at least a factor of 2 fewer function evaluations to achieve a given accuracy than a second order predictor–corrector method (center extrapolation / center midpoint method) with regards to Gd-157 concentration. Members of the EL family can be derived to at least fourth order. The EL3 and the EL4 algorithms presented are shown to be third and fourth order respectively on the systems of ODE test. In the Monte Carlo test, these methods did not overtake the accuracy of EPC methods before statistical uncertainty dominated the error. The statistical properties of the algorithms were also analyzed during the Monte Carlo problem. The new methods are shown to yield smaller standard deviations on final quantities as compared to the reference predictor–corrector method, by up to a factor of 1.4. [ABSTRACT FROM AUTHOR]

This paper introduces stochastic weighted particle algorithms for the solution of multi-compartment population balance equations. In particular, it presents a class of fragmentation weight transfer functions which are constructed such that the number of computational particles stays constant during fragmentation events. The weight transfer functions are constructed based on systems of weighted computational particles and each of it leads to a stochastic particle algorithm for the numerical treatment of population balance equations. Besides fragmentation, the algorithms also consider physical processes such as coagulation and the exchange of mass with the surroundings. The numerical properties of the algorithms are compared to the direct simulation algorithm and an existing method for the fragmentation of weighted particles. It is found that the new algorithms show better numerical performance over the two existing methods especially for systems with significant amount of large particles and high fragmentation rates. [ABSTRACT FROM AUTHOR]

• A linearly operator splitting algorithm is proposed for the nonlocal VCH equation. • The energy stabilities for both subproblems are proved. • The stability and convergence of the operator splitting algorithm are studied. • A semi-implicit SDC method is further used to improve time accuracy. Recently, the viscous Cahn-Hilliard (VCH) equation has been proposed as a phenomenological continuum model for phase separation in glass and polymer systems where intermolecular friction forces become important. Compared with the classical local VCH model, the nonlocal VCH model equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions of microstructures in materials. This paper presents a high order fast explicit method based on operator splitting and spectral deferred correction (SDC) for solving the nonlocal VCH equation. We start with a second-order operator splitting spectral scheme, which is based on the Fourier spectral method and the strong stability preserving Runge-Kutta (SSP-RK) method. The scheme takes advantage of applying the fast Fourier transform (FFT) and avoiding nonlinear iteration. The stability and convergence analysis of the obtained numerical scheme are analyzed. To improve the temporal accuracy, the semi-implicit SDC method is then introduced. Various numerical simulations are performed to validate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

• A new algorithm incorporates model reduction in the spatial and temporal domains. • We study parabolic problems with heterogeneous coefficients and rough initial data. • We derive convergence analysis that weakly depends on the heterogeneous coefficients. • The convergence is rigorously studied, which greatly improves the current result. • Extensive numerical tests are performed to show the fast convergence of our algorithm. We propose in this paper the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm to solve parabolic equations with heterogeneous coefficients efficiently. This algorithm combines the advantages of multiscale methods that can deal with heterogeneity in the spatial domain effectively, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. We derive the convergence rate of this algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step and the fine-scale time step. Extensive numerical tests are presented to demonstrate the performance of our algorithm, which verify our theoretical results perfectly. [ABSTRACT FROM AUTHOR]

• A neural network enhanced Boltzmann model is proposed. • The mechanical and neural models are unified into a differentiable architecture and the neural- ODE-type training strategy is constructed. • A general numerical scheme is designed to solve the universal Boltzmann equation. • Numerical experiments of homogeneous and inhomogeneous systems are provided to validate the current method. One of the biggest challenges for simulating the Boltzmann equation is the evaluation of fivefold collision integral. Given the recent successes of deep learning and the availability of efficient tools, it is an obvious idea to try to substitute the calculation of the collision operator by the evaluation of a neural network. However, it is unlcear whether this preserves key properties of the Boltzmann equation, such as conservation, invariances, the H-theorem, and fluid-dynamic limits. In this paper, we present an approach that guarantees the conservation properties and the correct fluid dynamic limit at leading order. The concept originates from a recently developed scientific machine learning strategy which has been named "universal differential equations". It proposes a hybridization that fuses the deep physical insights from classical Boltzmann modeling and the desirable computational efficiency from neural network surrogates. The construction of the method and the training strategy are demonstrated in detail. We conduct an asymptotic analysis and illustrate the multi-scale applicability of the method. The numerical algorithm for solving the neural network-enhanced Boltzmann equation is presented as well. Several numerical test cases are investigated. The results of numerical experiments show that the time-series modeling strategy enjoys the training efficiency on this supervised learning task. [ABSTRACT FROM AUTHOR]

• First dynamical low-rank algorithm that is mass, momentum, and energy conservative. • Can be combined with an explicit integrator that maintains conservation. • Conserves the underlying continuity equations in addition to the invariants. • Low-rank breaks the curse of dimensionality for high-dimensional kinetic equations. The primary challenge in solving kinetic equations, such as the Vlasov equation, is the high-dimensional phase space. In this context, dynamical low-rank approximations have emerged as a promising way to reduce the high computational cost imposed by such problems. However, a major disadvantage of this approach is that the physical structure of the underlying problem is not preserved. In this paper, we propose a dynamical low-rank algorithm that conserves mass, momentum, and energy as well as the corresponding continuity equations. We also show how this approach can be combined with a conservative time and space discretization. [ABSTRACT FROM AUTHOR]

Abstract: At the mesoscopic scale, chemical processes have probability distributions that evolve according to an infinite set of linear ordinary differential equations known as the chemical master equation (CME). Although only a few classes of CME problems are known to have exact and computationally tractable analytical solutions, the recently proposed finite state projection (FSP) technique provides a systematic reduction of the CME with guaranteed accuracy bounds. For many non-trivial systems, the original FSP technique has been shown to yield accurate approximations to the CME solution. Other systems may require a projection that is still too large to be solved efficiently; for these, the linearity of the FSP allows for many model reductions and computational techniques, which can increase the efficiency of the FSP method with little or no loss in accuracy. In this paper, we present a new approach for choosing and expanding the projection for the original FSP algorithm. Based upon this approach, we develop a new algorithm that exploits the linearity property of super-position. The new algorithm retains the full accuracy guarantees of the original FSP approach, but with significantly increased efficiency for some problems and a greater range of applicability. We illustrate the benefits of this algorithm on a simplified model of the heat shock mechanism in Escherichia coli. [Copyright &y& Elsevier]

Abstract: We present a new time-accurate algorithm for the explicit numerical integration of the compressible Euler equations of gas dynamics. This technique is based on the discrete-event simulation (DES) methodology for nonlinear flux-conservative PDEs [Y.A. Omelchenko, H. Karimabadi, Self-adaptive time integration of flux-conservative equations with sources, J. Comput. Phys. 216 (1) (2006) 179–194]. DES enables adaptive distribution of CPU resources in accordance with local time scales of the underlying numerical solution. It distinctly stands apart from multiple (local) time-stepping algorithms in that it requires neither selecting a global synchronization time step nor pre-determining a sequence of time-integration operations for individual parts of a heterogeneous numerical system. In this paper we extend the DES methodology in three important directions: (i) we apply DES to a system of coupled gas dynamics equations discretized via a central-upwind scheme [A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys. 160 (2000) 241–282; A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740]; (ii) we introduce a new Preemptive Event Processing (PEP) technique, which automatically enforces synchronous execution of events with sufficiently close update times; (iii) we significantly improve the accuracy of the previous algorithm [Y.A. Omelchenko, H. Karimabadi, Self-adaptive time integration of flux-conservative equations with sources, J. Comput. Phys. 216 (1) (2006) 179–194] by applying locally second-order-in-time flux-conserving corrections to the solution obtained with the forward Euler scheme. The performance of the new technique is demonstrated in a series of one-dimensional gas dynamics test problems by comparing numerical solutions obtained in event-driven and equivalent time-stepping simulations. [Copyright &y& Elsevier]

Abstract: In this paper, the exact discrete adjoint of an unstructured finite-volume formulation of the Euler equations in two dimensions is derived and implemented. The adjoint equations are solved with the same implicit scheme as used for the flow equations. The scheme is modified to efficiently account for multiple functionals simultaneously. An optimization framework, which couples an analytical shape parameterization to the flow/adjoint solver and to algorithms for constrained optimization, is tested on airfoil design cases involving transonic as well as supersonic flows. The effect of some approximations in the discrete adjoint, which aim at reducing the complexity of the implementation, is shown in terms of optimization results rather than only in terms of gradient accuracy. The shape-optimization method appears to be very efficient and robust. [Copyright &y& Elsevier]

Abstract: In this paper we develop an OcTree discretization for Maxwell’s equations in the quasi-static regime. We then use this discretization in order to develop a multigrid method for Maxwell’s equations with highly discontinuous coefficients. We test our algorithms and compare it to other multilevel algorithms. [Copyright &y& Elsevier]

Abstract: This paper presents an efficient adaptive mesh redistribution method to solve a non-linear Dirac (NLD) equation. Our algorithm is formed by three parts: the NLD evolution, the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh. At each time level, the equidistribution principle is first employed to iteratively redistribute coarse mesh points, and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity. After an adaptive coarse mesh is generated ideally and finally, each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain, and then the solution vector is remapped on the resulting new fine mesh by an affine method. The NLD equation is finally solved by using a high resolution shock-capturing method on the (fixed) non-uniform fine mesh. Extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the third-order rate of convergence, and yields an efficient and fast NLD solver that tracks and resolves both small, local and large solution gradients automatically. [Copyright &y& Elsevier]

Abstract: A numerical algorithm for kinetic modelling of droplet evaporation processes is suggested. This algorithm is focused on the direct numerical solution of the Boltzmann equations for two gas components: vapour and air. The physical and velocity spaces are discretised, and the Boltzmann equations are presented in discretised forms. The solution of these discretised equations is performed in two steps. Firstly, molecular displacements are calculated ignoring the effects of collisions. Secondly, the collisional relaxation is calculated under the assumption of spatial homogeneity. The conventional approach to calculating collisional integrals is replaced by the integration based on random cubature formulae. The distribution of molecular velocities after collisions is found based on the assumption that the total impulse and energy of colliding molecules are conserved. The directions of molecular impulses after the collisions are random, but the values of these impulses belong to an a priori chosen set. A new method of finding the matching condition for vapour mass fluxes at the outer boundary of the Knudsen layer of evaporating droplets and at the inner boundary of the hydrodynamic region is suggested. The numerical algorithm is applied to the analysis of three problems: the relaxation of an initially non-equilibrium distribution function towards the Maxwellian one, the analysis of the mixture of vapour and inert gas confined between two infinite plates and the evaporation of a diesel fuel droplet into a high pressure air. The solution of the second problem showed an agreement between the results predicted by the widely used Bird’s algorithm and the algorithm described in this paper. In the third problem the difference of masses and radii of vapour and air molecules is taken into account. The kinetic effects predicted by the numerical algorithm turned out to be noticeable if the contribution of air in the Knudsen layer is taken into account. [Copyright &y& Elsevier]

Abstract: In this paper, we describe in detail the computational algorithm used by our parallel multigrid elliptic equation solver with adaptive mesh refinement. Our code uses truncation error estimates to adaptively refine the grid as part of the solution process. The presentation includes a discussion of the orders of accuracy that we use for prolongation and restriction operators to ensure second order accurate results and to minimize computational work. Code tests are presented that confirm the overall second order accuracy and demonstrate the savings in computational resources provided by adaptive mesh refinement. [Copyright &y& Elsevier]

In this paper, we investigate the approximation of the solution to the Vlasov equation coupled with the Fokker–Planck–Landau collision operator using a phase space grid. On the one hand, the algorithm is based on the conservation of the flux of particles and the distribution function is reconstructed allowing to control spurious oscillations and preserving positivity and energy. On the other hand, the method preserves the main properties of the collision operators in order to reach the correct stationary state. Several numerical results are presented in one dimension in space and three dimensions in velocity. [Copyright &y& Elsevier]

In this paper, we develop a three-dimensional adaptive method based on iterative grid redistribution technique introduced in [J. Comput. Phys. 159 (2000) 246]. The key step for the successful implementation is a fast algorithm for grid generation, which is composed of solving the linear grid equation systems and an efficient method for inverting a map by computing iso-surface intersections. To carry out the three-dimensional calculations, the whole procedure is also parallelized on a PC cluster. The improved and extended three-dimensional adaptive method is applied to solve PDEs with singular solutions that have three-dimensional structures. Numerical experiments have demonstrated the method''s effectiveness. [Copyright &y& Elsevier]

The present paper highlights the importance of management of the discontinuous reconstruction in the kinetic schemes for gasdynamic equation systems. Firstly, it is revealed by the analysis of the gas kinetic-BGK scheme [JCP 171 (2001) 289] that a continuous reconstruction created from a discontinuous one is a key to the successful kinetic schemes. When it is applied to a well-resolved region, the numerical flux that takes account of the collision effect correctly becomes Lax–Wendroff-like. When it is applied to an unresolved region, such as a shock layer, an appreciable numerical dissipation, which contributes to the suppression of spurious oscillations, is produced. Secondly, new kinetic schemes for the compressible Navier–Stokes (Euler) equations are developed by using the key. The numerical flux of one of the schemes is computed by using the splitting algorithm, where the effect of the molecular collision is directly taken into account and the undesirable error of the splitting algorithm in the case where the time step is much larger than the mean free time is avoided by a simple modification of the initial data. Although a discontinuous reconstruction is employed in the approximation of the initial data, the continuity is automatically taken into account in the dominant part of the numerical flux. The other schemes are the extensions of the Lax–Wendroff-type scheme to the case of the key reconstruction. Thirdly, the performance of these schemes is tested. It is demonstrated that they work as shock capturing schemes and yield fine boundary-layer profiles with a reasonable number of cells, such as 10 cells in the layer. [Copyright &y& Elsevier]

This paper deals with the construction of algorithms which preserve the first integrals of a class of forced rigid motions, which retains a Hamiltonian structure, using Lie group methods such as those due to Lewis and Simo and Munthe-Kaas. For these mechanical systems, we also study the reconstruction of dynamical processes to describe the evolution of the orientation of the forced rigid body in space; for this we consider different algorithms to solve the kinematic equations. A comparison between these algorithms is made. Finally, we illustrate the numerical methods discussed here, studying the stabilization of the relative equilibrium corresponding to the stationary rotation about the intermediate axis in the presence of external torques about the minor axis. [Copyright &y& Elsevier]

A self-adjoint formulation of the energy transport model of semiconductor devices is proposed. This new formulation leads to symmetric and monotonic properties of the resulting system of nonlinear algebraic equations from an adaptive finite element approximation of the model. A node-by-node iterative method is then presented for solving the system. This is a globally convergent method that does not require the assembly of the global matrix system and full Jacobian matrices. An adaptive algorithm implementing this method is described in detail to illustrate the main features of this paper, namely, adaptation, node-by-node calculation, and global convergence. Numerical results of simulations on deep-submicron diode and MOSFET device structures are given to demonstrate the accuracy and efficiency of the algorithm. [Copyright &y& Elsevier]

• Parallel-in-time algorithms are developed for direct-adjoint loops. • Parallelization-in-time is feasible for linear and non-linear adjoint looping. • Theoretical scalings are available to assess the speedup a-priori. Parallel-in-time methods are developed to accelerate the direct-adjoint looping procedure. Particularly, we utilize the Paraexp algorithm, previously developed to integrate equations forward in time, to accelerate the direct-adjoint looping that arises from gradient-based optimization. We consider both linear and non-linear governing equations and exploit the linear, time-varying nature of the adjoint equations. Gains in efficiency are seen across all cases, showing that a Paraexp based parallel-in-time approach is feasible for the acceleration of direct-adjoint studies. This signifies a possible approach to further increase the run-time performance for optimization studies that either cannot be parallelized in space or are at their limit of efficiency gains for a parallel-in-space approach. Code demonstrating the algorithms considered in this paper is available. [ABSTRACT FROM AUTHOR]

• We propose a novel numerical algorithm for solving the Cahn-Hilliard-Navier-Stokes (CHNS) equations. • The scheme is second-order and linear while preserving the energy dissipation law. • The proposed scheme only requires solving linear systems, and the solution existence and uniqueness are guaranteed. • The proposed scheme obeys an energy dissipation law in the original variables. In this paper, we consider the Cahn-Hilliard equation coupled with the incompressible Navier-Stokes equation, usually known as the Cahn-Hilliard-Navier-Stokes (CHNS) system. The CHNS system has been widely embraced to investigate the dynamics of a binary fluid mixture. By utilizing the modified leap-frog time-marching method, we propose a novel numerical algorithm for solving the CHNS system in an efficient and accurate manner. This newly proposed scheme has several advantages. First of all, the proposed scheme is linear in time and space, such that only a linear algebraic system needs to be solved at each time-marching step, making it extremely efficient. Also, the existence and uniqueness of numerical solutions are guaranteed for any time step size. In addition, the scheme is unconditionally energy stable with second-order accuracy in time and spectral accuracy in space, such that relatively large temporal and spatial mesh sizes can be used to obtain reliable numerical solutions. The rigorous proofs for the unconditional energy stable property and solution existence and uniqueness are given. Furthermore, we present several numerical examples to test the proposed numerical algorithm and illustrate its accuracy and efficiency. The differences of coarsening dynamics between the Cahn-Hilliard equation and the Cahn-Hilliard-Navier-Stokes equations have been investigated as well. [ABSTRACT FROM AUTHOR]

In recent years the kinetic Fokker-Planck approach for modeling gas flows has become increasingly popular. In the Fokker-Planck ansatz the collision integral of the Boltzmann equation is approximated by a Fokker-Planck operator in velocity space. Instead of solving the resulting Fokker-Planck equation directly, the underlying random process is modeled, which leads to an efficient stochastic solution algorithm. Despite the attention to the Fokker-Planck ansatz, the modeling of polyatomic gases has been addressed only in a few works. In this paper a scheme is presented to extend arbitrary monatomic Fokker-Planck models to model polyatomic species. A master equation approach is used to model internal energy relaxation, but instead of solving the master equation directly, the underlying random process is simulated. Three different models are suggested to describe internal particle energies as continuous scalars or as a set of discrete energy levels. The proposed models are applied on different test cases to demonstrate their accuracy. Within the bounds of expectations, a very good agreement with reference DSMC simulations is achieved. • A Master equation approach is applied to model internal energy relaxation. • Three models of varying fidelity are constructed to describe internal energy states. • Prediction of vibrational energy levels consistent with the Larsen-Borgnakke model. • Test cases show very good agreement with reference DSMC simulations. [ABSTRACT FROM AUTHOR]