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• We proposed variable linear transformation improved physics-informed neural networks (VLT-PINNs) to solve thin-layer flow forward and inverse problems. • Three principles for determining the VLT parameters were identified for three types of variables in PINNs respectively. • The VLT-PINNs with the principle-suggested parameters show excellent performance over the ten test cases. • Only under the constraints of the VLT principles can we obtain satisfactory results, while traditional linear transformation may yield totally wrong results. • VLT method is an attempt to combine the three advantages of accuracy, universality and simplicity. Physics-informed neural networks (PINNs) have attracted wide attention due to their ability to seamlessly embed the learning process with physical laws and their considerable success in solving forward and inverse differential equation (DE) problems. While most studies are improving the learning process and network architecture of PINNs, less attention has been paid to the modification of the DE system, which may play an important role in addressing some limitations of PINNs. One of the simplest modifications that can be implemented to all DE systems is the variable linear transformation (VLT). Therefore, in this work, we propose the VLT-PINNs that solve the DE systems of the linear-transformed variables instead of the original ones. To clearly illustrate the importance of prior knowledge in determining the VLT parameters, we choose the thin-layer flow problems as our focus. Ten related cases were tested, including the jet flows, wake flows, mixing layers, boundary layers and Kovasznay flows. Based on the principle of normalization and for a better match of the DE system to the preference of NNs, we identify three principles for determining the VLT parameters: magnitude normalization for dependent variables (principle 1), local normalization for independent variables (principle 2), and appropriate scaling for physics-related parameters in inverse problems (principle 3). The VLT-PINNs with the VLT parameters suggested by the proposed principles show excellent performance over all the test cases, while the results are quite poor with the VLT parameters suggested by traditional linear transformations, such as nondimensionalization and global normalization. Comparison studies also show that only under the constraints of the VLT principles can we obtain satisfactory results. Besides, we find tanh is more appropriate as the activation function than sin for thin-layer flow problems, from both posteriori results and priori analyses with physical intuition. We highlight that our VLT method is an attempt to combine the three advantages of accuracy, universality and simplicity, and hope that it can provide new insights into the better integration of prior knowledge, physical intuition and the nature of NNs. The code for this paper is available on https://github.com/CAME-THU/VLT-PINN. [ABSTRACT FROM AUTHOR]

Projection methods are applicable in many fields. It is a natural and practical approach to devise the invariant-preserving schemes for conservative systems. The idea is to project the solution of any underlying numerical scheme onto the manifold determined by the invariant, and this process will be referred to as the projection integrator. Generally, the projection integrator chooses the gradient of invariant as its projection direction and has the same order as the underlying method. In this paper, we propose a different projection direction to construct a new projection integrator whose order is higher than the underlying method. According to this novel direction, we further summarize high-order projection integrators with superconvergence and rigorously prove the truncation error by utilizing the linear integral method as a central tool. Apart from the invariant-preserving property, symmetry is an important geometric property for reversible differential equations. The design and analysis of another high-order projection integrators with symmetry and superconvergence are also presented in this paper. Numerical experiments are provided to verify our theoretical results and illustrate that our proposed projection integrators have superior behaviors in a long time numerical simulation. [ABSTRACT FROM AUTHOR]

Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods. • Multi-dimensional summation-by-parts operators for general (non-polynomial) function spaces. • Existence proof that connects them to quadratures. • Construction procedure for general multi-dimensional domains/elements/blocks. • Proof of discrete conservation and energy-boundedness. • Numerical comparison demonstrating that non-polynomial function spaces can be of advantage. [ABSTRACT FROM AUTHOR]

In this paper, we propose a cusp-capturing physics-informed neural network (PINN) to solve discontinuous-coefficient elliptic interface problems whose solution is continuous but has discontinuous first derivatives on the interface. To find such a solution using neural network representation, we introduce a cusp-enforced level set function as an additional feature input to the network to retain the inherent solution properties; that is, capturing the solution cusps (where the derivatives are discontinuous) sharply. In addition, the proposed neural network has the advantage of being mesh-free, so it can easily handle problems in irregular domains. We train the network using the physics-informed framework in which the loss function comprises the residual of the differential equation together with certain interface and boundary conditions. We conduct a series of numerical experiments to demonstrate the effectiveness of the cusp-capturing technique and the accuracy of the present network model. Numerical results show that even using a one-hidden-layer (shallow) network with a moderate number of neurons and sufficient training data points, the present network model can achieve prediction accuracy comparable with traditional methods. Besides, if the solution is discontinuous across the interface, we can simply incorporate an additional supervised learning task for solution jump approximation into the present network without much difficulty. • Cusp-capturing physics informed neural network for elliptic interface problems. • Our network can present continuous solutions that inherently have discontinuous first derivatives on interfaces. • A mesh-free approach for solving PDEs with interfaces is presented. • Our network can achieve high prediction accuracy with large contrast discontinuous coefficients. [ABSTRACT FROM AUTHOR]

Abstract This article deals with the thermodynamic equilibrium recovery mechanisms in two-phase flows and their numerical modeling. The two phases, initially at different pressures, temperatures and chemical potentials, are supposed to be driven towards equilibrium conditions by three relaxation processes. First, a mechanical process applies to relax phasic pressures, then a thermal process, to allow the sensible heat transfer between the phases at different temperatures, and, lastly, a chemical process that is responsible for the mass transfer. The two-phase flow model is composed of six partial differential equations with source terms that allow the description of mixtures at full thermodynamic disequilibrium. Its homogeneous portion is hyperbolic and it is solved by a second-order accurate finite volume scheme that uses a HLLC-type approximate Riemann solver. The source terms modeling the relaxation processes are separately integrated as three systems of ordinary differential equations. The main contributions of this paper are: the capability of describing the possibly non-instantaneous time delay of equilibrium recover in a novel way, the equation of state independence of the numerical scheme, and the possibility to take into account the morphology of the flow pattern by using the interfacial area between phases. Highlights • Development of non-instantaneous relaxation techniques for the equilibrium recovery of the two-phase flows. • Equation of State (EoS) independence of the numerical techniques proposed. • Comparison of an accurate EoS to the Stiffened Gas one. • Analysis of the thermal and chemical disequilibrium using the IAPWS-IF97 equation of state. • Validation of the quasi-instantaneous simulations against the Homogeneous Equilibrium Model (HEM). [ABSTRACT FROM AUTHOR]

In this paper, we extend the implicit-explicit (IMEX) methods of Peer type recently developed by Lang and Hundsdorfer (2017) [16] to a broader class of two-step methods that allow the construction of super-convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes combine the necessary stability of implicit and low computational costs of explicit methods to efficiently solve systems of ordinary differential equations with both stiff and non-stiff parts included in the source term. To construct super-convergent IMEX-Peer methods with favourable stability properties, we derive necessary and sufficient conditions on the coefficient matrices and apply an extrapolation approach based on already computed stage values. Optimised super-convergent IMEX-Peer methods of order s + 1 for s = 2 , 3 , 4 stages are given as result of a search algorithm carefully designed to balance the size of the stability regions and the extrapolation errors. Numerical experiments and a comparison to other IMEX-Peer methods are included. [ABSTRACT FROM AUTHOR]

We consider in this paper the mathematical and numerical modeling of reflective boundary conditions (BC) associated to Boltzmann–Poisson systems, including diffusive reflection in addition to specularity, in the context of electron transport in semiconductor device modeling at nano scales, and their implementation in Discontinuous Galerkin (DG) schemes. We study these BC on the physical boundaries of the device and develop a numerical approximation to model an insulating boundary condition, or equivalently, a pointwise zero flux mathematical condition for the electron transport equation. Such condition balances the incident and reflective momentum flux at the microscopic level, pointwise at the boundary, in the case of a more general mixed reflection with momentum dependant specularity probability p ( k → ) . We compare the computational prediction of physical observables given by the numerical implementation of these different reflection conditions in our DG scheme for BP models, and observe that the diffusive condition influences the kinetic moments over the whole domain in position space. [ABSTRACT FROM AUTHOR]

• New high-order energy-preserving integrators are developed for Poisson systems. • Theoretical analysis including the existence and uniqueness, orders, implementation issues, and conservation of Casimir function is made. • The new integrators show much better numerical behaviour in long-term computation compared with the existing integrators. As is known, a system of differential equations possessing a first integral can be written in a non-canonical Hamiltonian system that often leads to a Poisson system under moderate conditions. Therefore, energy-preserving numerical integrators for such systems have become a subject of intensive investigation. In this paper, we design and analyse two classes of energy-preserving functionally-fitted integrators of arbitrary order for Poisson systems with highly oscillatory solutions using a new framework. We also show that both the order and the stage order of the proposed integrators may be affected by the used quadrature formula in practice. Furthermore, the existence and uniqueness of energy-preserving functionally-fitted integrators, their implementation issues, and the conservation of Casimir functions are investigated in detail. Finally, numerical experiments are carried out with the fourth-order and sixth-order energy-preserving integrators proposed in this paper, including the Lotka-Volterra system and the charged-particle dynamics in a strong magnetic field, and the numerical results demonstrate the remarkable accuracy and efficiency of our high-order energy-preserving integrators compared with the other energy-preserving methods in the literature. [ABSTRACT FROM AUTHOR]

The concept of physics-informed neural networks has become a useful tool for solving differential equations due to its flexibility. A few approaches use this concept to solve the eikonal equation that describes the first-arrival traveltimes of waves propagating in smooth heterogeneous velocity models. However, the challenge of the eikonal is exacerbated by the velocity models producing caustics, resulting in instabilities and deterioration of accuracy due to the non-smooth solution behavior. In this paper, we revisit the problem of solving the eikonal equation using neural networks to tackle caustic pathologies. We introduce the novel Neural Eikonal Solver (NES) for solving the isotropic eikonal equation in two formulations: the one-point problem is for a fixed source location; the two-point problem is for an arbitrary source-receiver pair. We present several techniques which provide stability in the case of caustics: improved factorization; non-symmetric loss function based on Hamiltonian; gaussian activation; symmetrization. In our tests, NES showed the relative mean-absolute error of 0.2-0.4% from the second-order factored Fast Marching Method with a similar inference time, and outperformed existing neural-network solvers giving 10-60 times lower errors and 2-30 times faster training. The one-point NES provides the most accurate solution, while the two-point NES gives slightly lower accuracy but an extremely compact representation with all spatial derivatives. It can be useful in many seismic problems: massive computations of traveltimes for millions of source-receiver pairs in Kirchhoff migration; modeling of ray amplitudes using spatial derivatives; traveltime tomography; earthquake localization; ray multipathing analysis. • Eikonal equation describes wave propagation in heterogeneous models. • Physics-informed neural networks can approximate the eikonal solution. • Physics-based modifications of the neural-network architecture enhance the accuracy. • Neural network compresses the solution, useful in large computational problems. • Neural network gives spatial derivatives, useful for calculation of wave amplitudes. [ABSTRACT FROM AUTHOR]

Exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations. Over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems. While the results of these investigations showed that exponential integrators can provide significant computational savings, the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages. Resistive magnetohydrodynamic (MHD) modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas. In many problems numerical solution of MHD equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest. In this paper we evaluate the performance of exponential integrators on large MHD problems and compare them to a state-of-the-art implicit time integrator. Both the variable and constant time step exponential methods of EPIRK-type are used to simulate magnetic reconnection and the Kevin–Helmholtz instability in plasma. Performance of these methods, which are part of the EPIC software package, is compared to the variable time step variable order BDF scheme included in the CVODE (part of SUNDIALS) library. We study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as Reynolds, Lundquist, and Prandtl numbers. We find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for MHD problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as MHD. [ABSTRACT FROM AUTHOR]

Abstract: This paper addresses some numerical and theoretical aspects of dual Schur domain decomposition methods for linear first-order transient partial differential equations. The spatially discrete system of equations resulting from a dual Schur domain decomposition method can be expressed as a system of differential-algebraic equations (DAEs). In this work, we consider the trapezoidal family of schemes for integrating the ordinary differential equations (ODEs) for each subdomain and present four different coupling methods, corresponding to different algebraic constraints, for enforcing kinematic continuity on the interface between the subdomains. Unlike the continuous formulation, the discretized formulation of the transient problem is unable to enforce simultaneously the continuity of both the primary variable and its rate along the subdomain interface (except for the backward Euler method). Method 1 (-continuity) is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid-point rule. To alleviate this difficulty, we propose a new Method 2 (modified -continuity) and prove its stability for coupling all time integrators in the trapezoidal family (except the forward Euler). Method 3 (-continuity) is based on enforcing the continuity of the time derivative of the primary variable. However, this constraint introduces a drift in the primary variable on the interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability. Our stability analysis is based on the “energy” method, and one of the main contributions of this paper is the extension of the energy method (which was previously introduced in the context of numerical methods for ODEs) to assess the stability of numerical formulations for index-2 differential-algebraic equations (DAEs). Finally, we present numerical examples to corroborate our theoretical predictions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the numerical approximation of the isothermal Euler equations for charged particles subject to the Lorentz force (the ‘Euler–Lorentz’ system). When the magnetic field is large, or equivalently, when the parameter representing the non-dimensional ion cyclotron frequency tends to zero, the so-called drift-fluid (or gyro-fluid) approximation is obtained. In this limit, the parallel motion relative to the magnetic field direction splits from perpendicular motion and is given implicitly by the constraint of zero total force along the magnetic field lines. In this paper, we provide a well-posed elliptic equation for the parallel velocity which in turn allows us to construct an Asymptotic-Preserving (AP) scheme for the Euler–Lorentz system. This scheme gives rise to both a consistent approximation of the Euler–Lorentz model when is finite and a consistent approximation of the drift limit when . Above all, it does not require any constraint on the space and time-steps related to the small value of . Numerical results are presented, which confirm the AP character of the scheme and its Asymptotic Stability. [Copyright &y& Elsevier]

Abstract: In this paper, a second-order scheme for the Quiet Direct Simulation (QDS) of Eulerian fluids is proposed. The QDS method replaces the random sampling method used in Direct Simulation Monte Carlo (DSMC) methods with a technique whereby particles are moved, have their properties distributed onto a mesh, are destroyed and then are recreated deterministically from the properties stored on the mesh using Gauss–Hermite quadrature weights and abscissas. Particles are permitted to move in physically realistic directions so flux exchange is not limited to cells sharing an adjacent interface as in conventional, direction decoupled finite volume solvers. In this paper the method is extended by calculating the fluxes of mass, momentum and energy between cells assuming a linear variation of density, temperature and velocity in each cell and using these fluxes to update the mass, velocity and internal energy carried by each particle. This Euler solver has several advantages including large dynamic range, no statistical scatter in the results, true direction fluxes to all nearby neighbors and is computationally inexpensive. The second-order method is found to reduce the numerical diffusion of QDS as demonstrated in several verification studies. These include unsteady shock tube flow, a two-dimensional blast wave and of the development of Mach 3 flow over a forward facing step in a wind tunnel, which are compared with previous results from the literature wherever is possible. Finally the implementation of QUIETWAVE, a rapid method of simulating blast events in urban environments, is introduced and the results of a test case are presented. [Copyright &y& Elsevier]

Abstract: This paper introduces the phase flow method, a novel, accurate and fast approach for constructing phase maps for nonlinear autonomous ordinary differential equations. The method operates by initially constructing the phase map for small times using a standard ODE integration rule and builds up the phase map for larger times with the help of a local interpolation scheme together with the group property of the phase flow. The computational complexity of building up the complete phase map is usually that of tracing a few rays. In addition, the phase flow method is provably and empirically very accurate. Once the phase map is available, integrating the ODE for initial conditions on the invariant manifold only makes use of local interpolation, thus having constant complexity. The paper develops applications in the field of high frequency wave propagation, and shows how to use the phase flow method to (1) rapidly propagate wave fronts, (2) rapidly calculate wave amplitudes along these wave fronts, and (3) rapidly evaluate multiple wave arrival times at arbitrary locations. [Copyright &y& Elsevier]

Abstract: This is the second paper of a series in which we build and study third-order Hermitian upwind schemes for numerically solving linear aeroacoustics. In this paper, we extend the method to non-stationary flows. For this purpose, we employ a family of six-wave models to discretize the space operator. This family is parametrized by an “acoustic propagation angle”. From this wave modelling, specific boundary conditions are proposed to treat effectively subsonic/supersonic boundary conditions. A sequence of numerical simulations is then carried out and makes it possible to examine the effectiveness of the scheme. [Copyright &y& Elsevier]

Abstract: In the recent paper by Dutt, Greengard and Rokhlin, a variant of deferred or defect correction methods is presented which couples Gaussian quadrature with the Picard integral equation formulation of the initial value ordinary differential equation. The resulting spectral deferred correction (SDC) methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy. In this paper, we show that for linear problems, the iterations in the SDC algorithm are equivalent to constructing a preconditioned Neumann series expansion for the solution of the standard collocation discretization of the ODE. This observation is used to accelerate the convergence of SDC using the GMRES Krylov subspace method. For nonlinear problems, the GMRES acceleration is coupled with a linear implicit approach. Stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy, efficiency, and stability of the original SDC approach. Furthermore, preliminary numerical experiments show that accelerating the convergence of SDC methods can effectively eliminate the order reduction previously observed for stiff ODE systems. [Copyright &y& Elsevier]

Abstract: A finite volume algorithm for the solution of the reaction–advection–diffusion equation on the sphere is derived and evaluated using analytical solutions. The proposed approach is based on the principle of semidiscretization. The convective and diffusive fluxes are approximated first, and then the resulting set of the ordinary differential equations (ODEs) is solved using the appropriate time stepping algorithm. In the first part of the paper, solutions to both the linear advection and the advection–diffusion problems for a single conservative scalar are discussed. The monotonicity of the scheme is achieved with the explicit adaptive dissipation. The development as well as the selected applications of the method are illustrated using a finite volume mesh constructed on the basis of geodesic icosahedral grid, which, in the past 40 years, has been frequently applied in different models of geophysical fluid dynamics. The performance of the solver is assessed using a suite of standard tests based on solid body rotation for different initial conditions. After analysis of the advection–diffusion problem, the extension of the method for the equations with reactive terms is presented. The performance of the solver is assessed by comparing the results to the analytical solution of the linearized reaction–diffusion system. In the final part of the paper, the application of the solver for studies of nonlinear reactions on the sphere is illustrated. The main intended application of the proposed method includes the simulation of transport of chemical constituents in the Earth’s atmosphere as well as the forecasting of moisture and cloud water fields in numerical weather prediction and climate models. [Copyright &y& Elsevier]

Abstract: Deriving high precision schemes to compute turbulent flows on fixed or moving complex grids is becoming a central issue in the direct numerical simulation (DNS) and large eddy simulation (LES) community. The step between classical DNS/LES codes on fixed structured grids and future methods on moving unstructured grids is a significant evolution in terms of numerical methods. For reacting flows, this evolution must also include more precise descriptions of multispecies flows and boundary conditions. This paper describes the development of a method for unsteady multispecies reacting flows on moving grids. The target field of application of this method is DNS and LES but this paper focuses on the method development and elementary test cases. The theoretical basis for the numerical method, the boundary conditions and the moving grid extension are first discussed. Various tests of the method are then provided on fixed and moving grids for simple reacting and non-reacting flows to demonstrate the precision and power of the method in simple reference laminar cases. [Copyright &y& Elsevier]

In this paper, a weak second-order accurate mid-point scheme for the stochastic differential equations (SDEs) arising in the composition PDF method for turbulent reactive flows is proposed and tested. The results are compared with two other schemes which are commonly used in the composition PDF method. In contrast to most higher-order schemes for SDEs, the present scheme uses a mid-point, which makes it especially suitable for the implementation of the position-advance fractional step in the composition PDF method. The scheme can also be applied to the PDF method used in conjunction with large eddy simulation (LES), since the SDEs considered in this paper include explicit time dependence of the drift and diffusion coefficients. Test calculations are reported, including a 2D unsteady case, to demonstrate the weak second-order accuracy of the scheme and to compare its performance to that of two first-order schemes. A new methodology for developing higher-order scheme for SDEs (by comparing the moments of the increments of the numerical approximation with those of the exact increments) is used to develop this scheme. [Copyright &y& Elsevier]

Mixed-precision algorithms combine low- and high-precision computations in order to benefit from the performance gains of reduced-precision without sacrificing accuracy. In this work, we design mixed-precision Runge–Kutta–Chebyshev (RKC) methods, where high precision is used for accuracy, and low precision for stability. Generally speaking, RKC methods are low-order explicit schemes with a stability domain growing quadratically with the number of function evaluations. For this reason, most of the computational effort is spent on stability rather than accuracy purposes. In this paper, we show that a naïve mixed-precision implementation of any Runge–Kutta scheme can harm the convergence order of the method and limit its accuracy, and we introduce a new class of mixed-precision RKC schemes that are instead unaffected by this limiting behavior. We present three mixed-precision schemes: a first- and a second-order RKC method, and a first-order multirate RKC scheme for multiscale problems. These schemes perform only the few function evaluations needed for accuracy (1 or 2 for first- and second-order methods respectively) in high precision, while the rest are performed in low precision. We prove that while these methods are essentially as cheap as their fully low-precision equivalent, they retain the stability and convergence order of their high-precision counterpart. Indeed, numerical experiments confirm that these schemes are as accurate as the corresponding high-precision method. • Mixed-precision algorithms can obtain high-accuracy results at a low-precision cost. • A naïve mixed-precision implementation can affect the convergence of ODE solvers. • A few high-precision function evaluations are sufficient to retain full accuracy. • Our mixed-precision Runge–Kutta–Chebyshev methods: fast and high-precision accurate. • We consider explicit stabilized methods for both single-scale and multirate problems. [ABSTRACT FROM AUTHOR]

The polynomial expansion method is a useful tool for solving both the direct and inverse Stokes problems, which together with the pointwise collocation technique is easy to derive the algebraic equations for satisfying the Stokes differential equations and the specified boundary conditions. In this paper we propose two novel numerical algorithms, based on a third–first order system and a third–third order system, to solve the direct and the inverse Cauchy problems in Stokes flows by developing a multiple-scale Pascal polynomial method, of which the scales are determined a priori by the collocation points. To assess the performance through numerical experiments, we find that the multiple-scale Pascal polynomial expansion method (MSPEM) is accurate and stable against large noise. [ABSTRACT FROM AUTHOR]

In this paper, we propose two new energy/dissipation-preserving Birkhoffian multi-symplectic methods (Birkhoffian and Birkhoffian box) for Maxwell's equations with dissipation terms. After investigating the non-autonomous and autonomous Birkhoffian formalism for Maxwell's equations with dissipation terms, we first apply a novel generating functional theory to the non-autonomous Birkhoffian formalism to propose our Birkhoffian scheme, and then implement a central box method to the autonomous Birkhoffian formalism to derive the Birkhoffian box scheme. We have obtained four formal local conservation laws and three formal energy global conservation laws. We have also proved that both of our derived schemes preserve the discrete version of the global/local conservation laws. Furthermore, the stability, dissipation and dispersion relations are also investigated for the schemes. Theoretical analysis shows that the schemes are unconditionally stable, dissipation-preserving for Maxwell's equations in a perfectly matched layer (PML) medium and have second order accuracy in both time and space. Numerical experiments for problems with exact theoretical results are given to demonstrate that the Birkhoffian multi-symplectic schemes are much more accurate in preserving energy than both the exponential finite-difference time-domain (FDTD) method and traditional Hamiltonian scheme. We also solve the electromagnetic pulse (EMP) propagation problem and the numerical results show that the Birkhoffian scheme recovers the magnitude of the current source and reaction history very well even after long time propagation. [ABSTRACT FROM AUTHOR]

This paper proposes an approach for high-order time integration within a multi-domain setting for time-fractional differential equations. Since the kernel is singular or nearly singular, two main difficulties arise after the domain decomposition: how to properly account for the history/memory part and how to perform the integration accurately. To address these issues, we propose a novel hybrid approach for the numerical integration based on the combination of three-term-recurrence relations of Jacobi polynomials and high-order Gauss quadrature. The different approximations used in the hybrid approach are justified theoretically and through numerical examples. Based on this, we propose a new multi-domain spectral method for high-order accurate time integrations and study its stability properties by identifying the method as a generalized linear method. Numerical experiments confirm hp -convergence for both time-fractional differential equations and time-fractional partial differential equations. [ABSTRACT FROM AUTHOR]

We observe that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In addition, we describe numerical experiments which illustrate several implications of this fact. For example, that many special functions of great interest — such as the Bessel functions J ν and Y ν — can be evaluated accurately using a number of operations which is O ( 1 ) in the order ν . The present paper is devoted to the development of an analytical apparatus. Numerical aspects of this work will be reported at a later date. [ABSTRACT FROM AUTHOR]

In this paper we consider multi-frequency highly oscillatory second-order differential equations x ″ ( t ) + M x ( t ) = f ( t , x ( t ) , x ′ ( t ) ) where high-frequency oscillations are generated by the linear part M x ( t ) , and M is positive semi-definite (not necessarily nonsingular). It is known that Filon-type methods are effective approach to numerically solving highly oscillatory problems. Unfortunately, however, existing Filon-type asymptotic methods fail to apply to the highly oscillatory second-order differential equations when M is singular. We study and propose an efficient improvement on the existing Filon-type asymptotic methods, so that the improved Filon-type asymptotic methods can be able to numerically solving this class of multi-frequency highly oscillatory systems with a singular matrix M . The improved Filon-type asymptotic methods are designed by combining Filon-type methods with the asymptotic methods based on the variation-of-constants formula. We also present one efficient and practical improved Filon-type asymptotic method which can be performed at lower cost. Accompanying numerical results show the remarkable efficiency. [ABSTRACT FROM AUTHOR]

Abstract: An exact expression of steady discrete shocks was recently obtained by the author in [9] for a class of residual-based compact schemes (RBC) applied to the inviscid Bürgers equation in a finite domain. Following the same lines, the analysis is extended to an infinite domain for a scalar conservation law with a general convex flux. For the dissipative high-order schemes considered, discrete shocks in infinite domain or with boundary conditions at short distance (Rankine–Hugoniot relations) are found to be very close. Besides, the present analytical description of shock capturing in infinite domain is explicit and so simple that it could lead to a new approach for correcting parasitic oscillations of high order RBC schemes. In a second part of the paper, exact solutions are also derived for equivalent differential equations (EDE) approximating schemes (subscript denotes the accuracy order) at orders 2p and . Although EDE involves Taylor expansions around steep structures, agreement between the exact EDE shock-profiles and the discrete shocks is remarkably good for and schemes. In addition, a strong similarity is demonstrated between the analytical expressions of discrete shocks and EDE shock profiles. [Copyright &y& Elsevier]

Abstract: In this paper, we present a simple and efficient computational algorithm for solving eigenvalue problems of high fractional-order differential equations with variable coefficients. The method of solution is based on utilizing the series solution to convert the governing fractional differential equation into a linear system of algebraic equations. Then, the eigenvalues can be calculated by finding the roots of the corresponding characteristic polynomial. Notice that this class of eigenvalue problems is very promising to the solution of linear fractional partial differential equations (FPDE). The numerical results demonstrate reliability and efficiency of the proposed algorithm. Based on our simulations some theoretical conjectures are reported. [Copyright &y& Elsevier]

• Spectral reconstructions are mighty techniques for removing the Gibbs phenomenon. • Most spectral reconstructions fail in the presence of truncated convolution errors. • Truncated convolution errors particularly appear in grating modes. • The Gegenbauer method is proved to remain exclusively robust against this phenomenon. Spectral reconstructions provide rigorous means to remove the Gibbs phenomenon and accelerate the convergence of spectral solutions in non-smooth differential equations. In this paper, we show the concurrent emergence of truncated convolution errors could entirely disrupt the performance of most reconstruction techniques in the vicinity of discontinuities. These errors arise when the Fourier coefficients of the product of two discontinuous functions, namely f = g h , are approximated via truncated convolution of the corresponding Fourier series, i.e. f ˆ k ≈ ∑ | ℓ | ⩽ N g ˆ ℓ h ˆ k − ℓ. Nonetheless, we numerically illustrate and rigorously prove that the classical Gegenbauer method remains exceptionally robust against this phenomenon, with the reconstruction error still diminishing proportional to O (N − 1) for the Fourier order N , and exponentially fast regardless of a constant. Finally, as a case study and a problem of interest in grating analysis whence the phenomenon initially was noticed, we demonstrate the emergence and practical resolution of truncated convolution errors in grating modes, which constitute the basis of Fourier modal methods. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper we presented a new ultra efficient numerical method for solving kinetic equations of the Boltzmann type (Dimarco and Loubere, 2013) [17]. The key idea, on which the method relies, is to solve the collision part on a grid and then to solve exactly the transport part by following the characteristics backward in time. On the contrary to classical semi-Lagrangian methods one does not need to reconstruct the distribution function at each time step. This allows to tremendously reduce the computational cost and to perform efficient numerical simulations of kinetic equations up to the six-dimensional case without parallelization. However, the main drawback of the method developed was the loss of spatial accuracy close to the fluid limit. In the present work, we modify the scheme in such a way that it is able to preserve the high order spatial accuracy for extremely rarefied and fluid regimes. In particular, in the fluid limit, the method automatically degenerates into a high order method for the compressible Euler equations. Numerical examples are presented which validate the method, show the higher accuracy with respect to the previous approach and measure its efficiency with respect to well-known schemes (Direct Simulation Monte Carlo, Finite Volume, MUSCL, WENO). [Copyright &y& Elsevier]

Abstract: In this paper we present a new ultra efficient numerical method for solving kinetic equations. In this preliminary work, we present the scheme in the case of the BGK relaxation operator. The scheme, being based on a splitting technique between transport and collision, can be easily extended to other collisional operators as the Boltzmann collision integral or to other kinetic equations such as the Vlasov equation. The key idea, on which the method relies, is to solve the collision part on a grid and then to solve exactly the transport linear part by following the characteristics backward in time. The main difference between the method proposed and semi-Lagrangian methods is that here we do not need to reconstruct the distribution function at each time step. This allows to tremendously reduce the computational cost of the method and it permits for the first time, to the authorʼs knowledge, to compute solutions of full six dimensional kinetic equations on a single processor laptop machine. Numerical examples, up to the full three dimensional case, are presented which validate the method and assess its efficiency in 1D, 2D and 3D. [Copyright &y& Elsevier]

Fractional Ginzburg-Landau equations as generalizations of the classical one have been used to describe various physical phenomena. In this paper, we propose a numerical integration method for space fractional Ginzburg-Landau equations based on a dynamical low-rank approximation. We first approximate the space fractional derivatives by using a fractional centered difference method. Then, the resulting matrix differential equation is split into a stiff linear part and a nonstiff (nonlinear) one. For solving these two subproblems, a dynamical low-rank approach is employed. The convergence of our method is proved rigorously. Numerical examples are reported which show that the proposed method is robust and accurate. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the numerical modeling of transient mechanical waves in linear viscoelastic solids. Dissipation mechanisms are described using the generalized Zener model. No time convolutions are required thanks to the introduction of memory variables that satisfy local-in-time differential equations. By appropriately choosing the relaxation parameters, it is possible to accurately describe a large range of materials, such as solids with constant quality factors. The evolution equations satisfied by the velocity, the stress, and the memory variables are written in the form of a first-order system of PDEs with a source term. This system is solved by splitting it into two parts: the propagative part is discretized explicitly, using a fourth-order ADER scheme on a Cartesian grid, and the diffusive part is then solved exactly. Jump conditions along the interfaces are discretized by applying an immersed interface method. Numerical experiments of wave propagation in viscoelastic and fluid media show the efficiency of this numerical modeling for dealing with challenging problems, such as multiple scattering configurations. [Copyright &y& Elsevier]

Abstract: Nowadays, fractional calculus are used to model various different phenomena in nature, but due to the non-local property of the fractional derivative, it still remains a lot of improvements in the present numerical approaches. In this paper, some new numerical approaches based on piecewise interpolation for fractional calculus, and some new improved approaches based on the Simpson method for the fractional differential equations are proposed. We use higher order piecewise interpolation polynomial to approximate the fractional integral and fractional derivatives, and use the Simpson method to design a higher order algorithm for the fractional differential equations. Error analyses and stability analyses are also given, and the numerical results show that these constructed numerical approaches are efficient. [Copyright &y& Elsevier]

Abstract: The meshless local Petrov–Galerkin (MLPG) method in conjunction with the modified precise time step integration method in the time domain is proposed for transient heat conduction analysis in this paper. The MLPG method is often referred to as a truly meshless method because it requires no elements or background cells for either field interpolation or background integration. Local weak forms are developed using weighted residual method locally from the partial differential equation of transient heat conduction. In order to simplify the treatment of essential boundary conditions, the natural neighbour interpolation (NNI) is employed for the construction of trial functions. Moreover, the three-node triangular FEM shape functions are taken as test functions to reduce the order of integrands involved in domain integrals. The semi-discrete heat conduction equation is solved numerically with modified precise time step integration method in the time domain. The availability and accuracy of the present method for transient heat conduction analysis are tested through numerical examples. [Copyright &y& Elsevier]

Abstract: Satellite-observed radiance is a nonlinear functional of surface properties and atmospheric temperature and absorbing gas profiles as described by the radiative transfer equation (RTE). In the era of hyperspectral sounders with thousands of high-resolution channels, the computation of the radiative transfer model becomes more time-consuming. The radiative transfer model performance in operational numerical weather prediction systems still limits the number of channels we can use in hyperspectral sounders to only a few hundreds. To take the full advantage of such high-resolution infrared observations, a computationally efficient radiative transfer model is needed to facilitate satellite data assimilation. In recent years the programmable commodity graphics processing unit (GPU) has evolved into a highly parallel, multi-threaded, many-core processor with tremendous computational speed and very high memory bandwidth. The radiative transfer model is very suitable for the GPU implementation to take advantage of the hardware’s efficiency and parallelism where radiances of many channels can be calculated in parallel in GPUs. In this paper, we develop a GPU-based high-performance radiative transfer model for the Infrared Atmospheric Sounding Interferometer (IASI) launched in 2006 onboard the first European meteorological polar-orbiting satellites, METOP-A. Each IASI spectrum has 8461 spectral channels. The IASI radiative transfer model consists of three modules. The first module for computing the regression predictors takes less than 0.004% of CPU time, while the second module for transmittance computation and the third module for radiance computation take approximately 92.5% and 7.5%, respectively. Our GPU-based IASI radiative transfer model is developed to run on a low-cost personal supercomputer with four GPUs with total 960 compute cores, delivering near 4 TFlops theoretical peak performance. By massively parallelizing the second and third modules, we reached 364× speedup for 1GPU and 1455× speedup for all 4GPUs, both with respect to the original CPU-based single-threaded Fortran code with the –O2 compiling optimization. The significant 1455× speedup using a computer with four GPUs means that the proposed GPU-based high-performance forward model is able to compute one day’s amount of 1,296,000 IASI spectra within nearly 10min, whereas the original single CPU-based version will impractically take more than 10days. This model runs over 80% of the theoretical memory bandwidth with asynchronous data transfer. A novel CPU–GPU pipeline implementation of the IASI radiative transfer model is proposed. The GPU-based high-performance IASI radiative transfer model is suitable for the assimilation of the IASI radiance observations into the operational numerical weather forecast model. [Copyright &y& Elsevier]

Abstract: This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent–solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the solvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By optimizing the total free energy functional, we derive coupled generalized Poisson–Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent–solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent–solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second-order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature. [Copyright &y& Elsevier]

Abstract: We investigate the traditional kinetic flux vector splitting (KFVS) and BGK schemes for the compressible Euler equations. First, based on a careful study of the behavior of the discrete physical variables across the contact discontinuity, we analyze quantitatively the mechanism of inducing spurious oscillations of the velocity and pressure in the vicinity of the contact discontinuity for the first-order KFVS and BGK schemes. Then, with the help of this analysis, we propose a first-order modified KFVS (MKFVS) scheme which is oscillation-free in the vicinity of the contact discontinuity, provided certain consistent conditions are satisfied. Moreover, by using piecewise linear reconstruction and van Leer’s limiter, the first-order MKFVS scheme is extended to a second-order one, consequently, a nonoscillatory second-order MKFVS scheme is constructed. Finally, by combing the MKFVS schemes with the γ-model, we successfully extend the MKFVS schemes to multi-flows, and propose therefore a first- and second-order MKFVS schemes for multi-fluid computations, which are nonoscillatory across fluid interfaces. A number of numerical examples presented in this paper validate the theoretic analysis and demonstrate the good performance of the MKFVS schemes in simulation of contact discontinuities for both single- and multi-fluids. [Copyright &y& Elsevier]

Abstract: It is well known, that standard upwind schemes for the Euler equations face a number of problems in the low Mach number regime: stiffness, cancellation and accuracy problems. A new aspect, presented in this paper, is the dependence on the cell geometry: applied on a triangular grid, the accuracy problem disappears, i.e. flows of arbitrarily small Mach numbers can be simulated on a fixed mesh. We give an asymptotic analysis of this, up to date unknown, phenomenon for the first-order Roe scheme and present a number of numerical results. [Copyright &y& Elsevier]

Abstract: The analysis of the time-splitting procedure implemented in the Princeton Ocean Model (POM) is presented. The time-splitting procedure uses different time steps to describe the evolution of interacting fast and slow propagating modes. In the general case the exact separation of the fast and slow modes is not possible. The main idea of the analyzed procedure is to split the system of primitive equations into two systems of equations for interacting external and internal modes. By definition, the internal mode varies slowly and the crux of the problem is to determine the proper filter, which excludes the fast component of the external mode variables in the relevant equations. The objective of this paper is to examine properties of the POM time-splitting procedure applied to equations governing the simplest linear non-rotating two-layer model of constant depth. The simplicity of the model makes it possible to study these properties analytically. First, the time-split system of differential equations is examined for two types of the determination of the slow component based on an asymptotic approach or time-averaging. Second, the differential-difference scheme is developed and some criteria of its stability are discussed for centered, forward, or backward time-averaging of the external mode variables. Finally, the stability of the POM time-splitting schemes with centered and forward time-averaging is analyzed. The effect of the Asselin filter on solutions of the considered schemes is studied. It is assumed that questions arising in the analysis of the simplest model are inherent in the general model as well. [Copyright &y& Elsevier]

Abstract: In this paper, we report a new development of image charge approximations of reaction fields for a charge inside a dielectric spherical cavity immersed in an ionic solvent with arbitrary ionic strength. This new development removes the requirement of low ionic strength of the solvent in a previous result [S. Deng, W. Cai, Extending the fast multipole method for charges inside a dielectric sphere in an ionic solvent: high-order image approximations for reaction fields, J. Comput. Phys. 227 (2007) 1246–1266], thus extending the applicability of the image charge approximations of reaction fields in the modeling of biomolecular solvation. [Copyright &y& Elsevier]

• 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]

• A general framework to design arbitrarily high order energy-preserving exponential integrators is presented. • The fourth-order energy-preserving exponential integrator is constructed and tested in detail. • Numerical results demonstrate the high efficiency and good energy preservation of the proposed integrator. Taking into account the limited accuracy of the energy-preserving exponential integrator of order two (Li and Wu, 2016 [29]) for conservative or dissipative systems with highly oscillatory solutions, this paper is devoted to presenting a uniform framework to design energy-preserving exponential integrators of arbitrarily high order based on the modifying integrator theory. To this end, we first show that the second-order energy-preserving exponential integrator is a B-series method. Using the adapted substitution law, we then prove that there exist arbitrary order energy-preserving exponential integrators and show how to design arbitrarily high-order integrators by finding the truncated modified differential equations. As an example, the fourth-order energy-preserving exponential integrator is constructed in detail. The stability and convergence of the proposed integrators are analyzed as well. Finally, numerical experiments are accompanied, including both ODEs and PDEs, and the numerical results demonstrate the remarkable superiority over the existing energy-preserving integrators for highly oscillatory systems in the literature. [ABSTRACT FROM AUTHOR]

Abstract: The numerical solution of the one-dimensional nonlinear Klein-Gordon equation on an unbounded domain is studied in this paper. Split local absorbing boundary (SLAB) conditions are obtained by the operator splitting method, then the original problem is reduced to an initial boundary value problem on a bounded computational domain, which can be solved by the finite difference method. Several numerical examples are provided to show the advantages and effectiveness of the given method, and some interesting collision behaviors are also observed. [Copyright &y& Elsevier]

Abstract: This paper presents and compares various numerical techniques for the long-wave short-wave interaction equations. In addition to the standard explicit, implicit schemes and the spectral methods, a novel scheme SRK which is based on a time-splitting approach combined with the Runge–Kutta method is presented. We demonstrate that not only the SRK scheme is efficient compared to the split step spectral methods, but it can apply directly to problems with general boundary conditions. The conservation properties of the numerical schemes are discussed. Numerical simulations are reported for case studies with different types of initial data. The present study enhances our understanding of the behavior of nonlinear dispersive waves in the semi-classical limit. [Copyright &y& Elsevier]

Abstract: This paper presents a novel methodology for simulating incompressible two-phase flows by combining an improved version of the conservative level set technique introduced in [E. Olsson, G. Kreiss, A conservative level set method for two phase flow, J. Comput. Phys. 210 (2005) 225–246] with a ghost fluid approach. By employing a hyperbolic tangent level set function that is transported and re-initialized using fully conservative numerical schemes, mass conservation issues that are known to affect level set methods are greatly reduced. In order to improve the accuracy of the conservative level set method, high order numerical schemes are used. The overall robustness of the numerical approach is increased by computing the interface normals from a signed distance function reconstructed from the hyperbolic tangent level set by a fast marching method. The convergence of the curvature calculation is ensured by using a least squares reconstruction. The ghost fluid technique provides a way of handling the interfacial forces and large density jumps associated with two-phase flows with good accuracy, while avoiding artificial spreading of the interface. Since the proposed approach relies on partial differential equations, its implementation is straightforward in all coordinate systems, and it benefits from high parallel efficiency. The robustness and efficiency of the approach is further improved by using implicit schemes for the interface transport and re-initialization equations, as well as for the momentum solver. The performance of the method is assessed through both classical level set transport tests and simple two-phase flow examples including topology changes. It is then applied to simulate turbulent atomization of a liquid Diesel jet at . The conservation errors associated with the accurate conservative level set technique are shown to remain small even for this complex case. [Copyright &y& Elsevier]

Abstract: A global description of discontinuous functions is introduced in this paper. By expressing a discontinuous function as the sum of a smooth function and a correction term determined by jump conditions, we turn the unknown function from a discontinuous one into a sufficiently smooth one when solving a differential equation. Spectral schemes are developed based on this concept with the intention of eliminating or reducing the Gibbs oscillation. Finite difference schemes are also constructed as an alternative of the current immersed interface methods. Both spectral and finite difference schemes are tested on one- and two-dimensional cases. [Copyright &y& Elsevier]

Abstract: This paper is devoted to comparing numerical schemes for a differential equation with convection and fourth-order diffusion. Our model equation is , which arises in the context of thin film flow. First we employ implicit schemes and treat both convection and diffusion terms implicitly. Then the convection terms are treated with well-known explicit schemes, namely Godunov, WENO and an upwind-type scheme, while the diffusion term is still treated implicitly. The diffusion and convection schemes are combined using a fractional step-splitting method. [Copyright &y& Elsevier]

Abstract: A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t 1/2, yielding a convergence rate of 1/2. Self-similar heuristics for higher-order discretizations produce a growth rate for the layer thickness of Δt 1/(p+1) which yields an estimate for the convergence rate as p/(p +1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations. [Copyright &y& Elsevier]

Abstract: Non-equilibrium Green’s function (NEGF) is a general method for modeling non-equilibrium quantum transport in open mesoscopic systems with many body scattering effects. In this paper, we present a unified treatment of quantum device boundaries in the framework of NEGF with both finite difference and finite element discretizations. Boundary treatments for both types of numerical methods, and the resulting self-energy Σ for the NEGF formulism, representing the dissipative effects of device contacts on the transport, are derived using auxiliary Green’s functions for the exterior of the quantum devices. Numerical results with both discretization schemes for an one-dimensional nano-device and a 29nm double gated MOSFET are provided to demonstrate the accuracy and flexibility of the proposed boundary treatments. [Copyright &y& Elsevier]

Abstract: A discrete velocity direction model for the Boltzmann equation is proposed in this paper, which provides an alternative technique to the rarefied gas flows. In this model, the directions of molecular velocities are discrete, which are restricted in eight fixed directions, while the molecular speed rate is still continuous. By this approximation, the Boltzmann equation in the six-dimensional phase space is replaced by eight differential-integral equations in three-dimensional space. Thus, the computational cost is reduced greatly by reduction of three dimensions. The number of discrete velocities is not fixed in the present model because the speed rate can be truncated arbitrarily. This is distinguished from the conventional discrete velocity models (DVM). To test this technique, it was applied to the Couette flow and Poiseuille flow. The computed results agree well with those by the linearized Boltzmann equation and the DSMC method. [Copyright &y& Elsevier]