First-order Scheme Research Articles

Nonsmooth dynamic analysis of rigid-flexible interaction collision

This paper aims to explore the deformation of the collided bodies in multi-body systems and to effectively simulate the motion path of colliding bodies. First, we describe the geometrically nonlinear problems of materials by the total Lagrangian formulation. Second, a first-order integration scheme is used to solve the dynamics equations. An algorithm combining the bi-potential method with the node-to-point contact identification is proposed to solve the interface problems of rigid-flexible interaction collision. To observe the collision process more intuitively, the internal software FER/VIEW is introduced to visualize the results. The accuracy is proved by comparing the proposed method with the analytical solution or another numerical solution. Moreover, the proposed method has more numerical robustness, such as occupying less computer storage, saving the computational cost, and broadening the application range of the bi-potential method.

Kesirli Mertebeden Pseudo Hiperbolik Diferansiyel Denklemlerin Sonlu Fark Metodu ile Nümerik Çözümleri

Fractional differential equations are useful for modelling mathematical issues in fields including engineering, physics, and biology. In this article, approximate solutions of an initial boundary value problem for fractional pseudo hyperbolic differential equations are investigated using the finite difference method. First, a first-order finite difference scheme is created for the initial boundary value problem. Then, stability analysis was performed for this finite difference scheme. In order to support the theoretical results obtained, error values were calculated for precise and approximate solutions in different fractional order derivatives of a sample problem. Some numerical simulations are also given to show the effectiveness of the applied solution method.

Analysis of divergence-free 𝐻¹ conforming FEM with IMEX-SAV scheme for the Navier-Stokes equations at high Reynolds number

In this paper, we analyze the first-order implicit-explicit type scheme based on the scalar auxiliary variable (SAV) with divergence-free H 1 H^1 conforming finite element method (FEM) in space for the evolutionary incompressible Navier-Stokes equations at high Reynolds number. The stability and a priori error estimates are given, in which the constants are independent of the Reynolds number. The velocity energy estimate is given without any condition on the time step, however, the a priori error estimates for the velocity are obtained with severe time step restrictions. In addition, a Reynolds-dependent error bound with convergence order of k + 1 k+1 in space is also obtained for the velocity error in the L 2 L^2 norm with no time step restrictions. Here, k k is the polynomial order of the velocity space. Some numerical experiments are carried out to verify the analytical results.

Solving the Incompressible Surface Stokes Equation by Standard Velocity-Correction Projection Methods

In this paper, an effective numerical algorithm for the Stokes equation of a curved surface is presented and analyzed. The velocity field was decoupled from the pressure by the standard velocity correction projection method, and the penalty term was introduced to make the velocity satisfy the tangential condition. The first-order backward Euler scheme and second-order BDF scheme are used to discretize the time separately, and the stability of the two schemes is analyzed. The mixed finite element pair (P2,P1) is applied to discretization of space. Finally, numerical examples are given to verify the accuracy and effectiveness of the proposed method.

An upwind generalized finite difference method (GFDM) for meshless analysis of heat and mass transfer in porous media

In this paper, an upwind GFDM is developed for coupled heat and mass transfer problems in porous media. GFDM is a meshless method that can obtain the difference schemes of spatial derivatives by using Taylor expansion in local node influence domains and the weighted least squares method. The first-order single-point upstream scheme in the FDM/FVM-based reservoir simulator is introduced to GFDM to form the upwind GFDM, based on which a sequential coupled discrete scheme of the pressure diffusion equation and the heat convection-conduction equation is solved to obtain pressure and temperature profiles. This paper demonstrates that this method can be used to obtain the meshless solution of the convection–diffusion equation with a stable upwind effect. For porous flow problems, the upwind GFDM is more practical and stable than the method of manually adjusting the influence domain based on the prior information of the flow field to achieve the upwind effect. Two types of calculation errors are analyzed, and three numerical examples are implemented to illustrate the good calculation accuracy and convergence of the upwind GFDM for heat and mass transfer problems in porous media and indicate the increase in the radius of the node influence domain will increase the calculation error of temperature profiles. Overall, the upwind GFDM discretizes the computational domain using only a point cloud that is generated with much less topological constraints than the generated mesh, but achieves good computational performance as the mesh-based approaches, and therefore has great potential to be developed as a general-purpose numerical simulator for various porous flow problems in domains with complex geometry.

Highly Accurate Compact Finite Difference Schemes for Two-Point Boundary Value Problems with Robin Boundary Conditions

In this study, a high-order compact finite difference method is used to solve boundary value problems with Robin boundary conditions. The norm is to use a first-order finite difference scheme to approximate Neumann and Robin boundary conditions, but that compromises the accuracy of the entire scheme. As a result, new higher-order finite difference schemes for approximating Robin boundary conditions are developed in this work. Six examples for testing the applicability and performance of the method are considered. Convergence analysis is provided, and it is consistent with the numerical results. The results are compared with the exact solutions and published results from other methods. The method produces highly accurate results, which are displayed in tables and graphs.

Improvement of the vertical mixing in chemistry transport modeling based on a 1.5-order turbulence kinetic energy-based eddy diffusivity closure scheme

In this study, firstly, three first-order local closure vertical diffusion schemes used in multiple mesoscale chemical transport models (CTM) were embedded in the CHIMERE CTM model and tested over a one-year simulation covering the whole France. Three model configurations present a fair reproduction of pollutant concentrations both in urban and rural areas, indicating it is an effective way to reproduce the dispersion of pollutants in chemistry transport modeling. However, it cannot be expected to significantly improve the vertical mixing under the first-order closure scheme. Secondly, a 1.5-order turbulence kinetic energy-based eddy diffusivity closure scheme called as the new eddy diffusion (NED) is implemented in CHIMERE to describe more realistic diffusion processes near the surface. A fifteen-day simulation encompassing a winter pollution episode was performed for three major cities with horizontal resolution of 1.67 km and the first layer height at 12 m, respectively. The NED scheme improved NO2 simulations at most urban sites compared to the initial Kz diffusion scheme (IKD). Taking the root mean square error as evaluation criteria, the average improvements are 18.8%, 24.5% and 9.5% for NO2 simualtion in Paris, Lyon and Bordeaux respectively. For the model performance of PM2.5 and PM10 simulations in the urban areas of Paris, the improvements are 13.5% and 19.1%, respectively. Overall, preliminary outcomes of this study are encouraging. The simulation with more sophisticated and realistic eddy viscosities are better than for IKD that is widely used in CTMs, but we need to realize that this is only a fifteen-day simulation for three cities, for further research, longer periods are needed with a greater variety of meteorological situations to prove the universality of the NED scheme.

Simulation of multidimensional diffusions with sticky boundaries via Markov chain approximation

In this study, we develop a novel method to simulate multidimensional diffusions with sticky boundaries for which the Euler scheme fails. We approximate the sticky diffusion process by a multidimensional continuous-time Markov chain (CTMC) that can be easily simulated. We provide two approaches to construct the CTMC. In the first approach, we approximate the infinitesimal generator of the sticky diffusion by finite difference using standard coordinate directions. In the second one, we match the local moments using the drift and the eigenvectors of the covariance matrix as transition directions. The first approach does not always guarantee a valid Markov chain, whereas the second one can. We show that both construction methods yield a first-order simulation scheme, which can capture the sticky behavior and is free from the curse of dimensionality. As applications, we use our method to simulate the limit of a queuing system with exceptional service policy and a multi-factor short rate model for low interest rate environment.

Frequency-Explicit A Posteriori Error Estimates for Finite Element Discretizations of Maxwell's Equations

We consider residual-based a posteriori error estimators for Galerkin-type discretizations of time-harmonic Maxwell's equations. We focus on configurations where the frequency is high, or close to a resonance frequency, and derive reliability and efficiency estimates. In contrast to previous related works, our estimates are frequency-explicit. In particular, our key contribution is to show that even if the constants appearing in the reliability and efficiency estimates may blow up on coarse meshes, they become independent of the frequency for sufficiently refined meshes. Such results were previously known for the Helmholtz equation describing scalar wave propagation problems and we show that they naturally extend, at the price of many technicalities in the proofs, to Maxwell's equations. Our mathematical analysis is performed in the 3D case, and covers conforming N\'ed\'elec discretizations of the first and second family, as well as first-order (and hybridizable) discontinuous Galerkin schemes. We also present numerical experiments in the 2D case, where Maxwell's equations are discretized with N\'ed\'elec elements of the first family. These illustrating examples perfectly fit our key theoretical findings, and suggest that our estimates are sharp.

TVD-MOOD schemes based on implicit-explicit time integration

The context of this work is the development of first order total variation diminishing (TVD) implicit-explicit (IMEX) Runge-Kutta (RK) schemes as a basis of a Multidimensional Optimal Order detection (MOOD) approach to approximate the solution of hyperbolic multi-scale equations. A key feature of our newly proposed TVD schemes is that the resulting CFL condition does not depend on the fast waves of the considered model, as long as they are integrated implicitly. However, a result from Gottlieb et al. [1] gives a first order barrier for unconditionally stable implicit TVD-RK schemes and TVD-IMEX-RK schemes with scale-independent CFL conditions. Therefore, the goal of this work is to consistently improve the resolution of a first-order IMEX-RK scheme, while retaining its L∞ stability and TVD properties. In this work we present a novel approach based on a convex combination between a first-order TVD IMEX Euler scheme and a potentially oscillatory high-order IMEX-RK scheme. We derive and analyse the TVD property for a scalar multi-scale equation and numerically assess the performance of our TVD schemes compared to standard L-stable and SSP IMEX RK schemes from the literature. Finally, the resulting TVD-MOOD schemes are applied to the isentropic Euler equations.

Error estimates for the Scalar Auxiliary Variable (SAV) schemes to the modified phase field crystal equation

We design first-order and second-order time-stepping schemes for the modified phase field crystal model based on the scalar auxiliary variable method in this work. The model is a nonlinear sixth-order damped wave equation that includes both elastic interactions and diffusive dynamics. Our schemes are linear and satisfy the unconditional energy stability with respect to pseudo energy. We also rigorously estimate the errors of the numerical schemes. Finally, some numerical tests are presented to validate our theoretical results.

On the Discretization of Richards Equation in Canadian Land Surface Models

ABSTRACT Both the Canadian Land Surface Scheme (CLASS) and the Soil, Vegetation, and Snow (SVS) land surface models employ a potentially problematic discretization of Richards equation for unsaturated vertical flow in the soil column. It is shown here that this discretization will always overestimate the vertical moisture gradient compared to a better-constructed first-order scheme, which under some circumstances could lead to erroneous moisture drawdown. The problem stems from an interpolation calculation on the irregularly spaced grids traditionally used by these models. While vanishing on uniform grids, this numerical error progressively worsens with increasing layer thickness differences. In this brief technical note a systematic method for developing first and higher order schemes on irregular, staggered grids is presented. To demonstrate the potential impact of the new first – order scheme, multi-year simulations of five FLUXNET sites are presented and discussed. A dramatic improvement in first layer soil moisture is found for two of the sites, which contributes to potentially significant differences in evapotranspiration. Higher order schemes are also possible but must be constructed carefully, in concert with a judicious choice of soil layer spacing in order to minimize discretization error. Given the extensive use of CLASS and SVS in Canadian environmental prediction systems, and the freedom with which users can specify soil layer thicknesses, it is recommended that modellers consider this issue carefully in their applications.

High-order well-balanced and positivity-preserving finite-difference AWENO scheme with hydrostatic reconstruction for shallow water equations

The shallow water equations (SWEs) admit still water steady-state solutions in which the flux gradients are exactly balanced by the source term. Furthermore, the no-water dry areas make the numerical simulation of the SWEs more difficult since the water height often loses positivity near the dry areas. In this study, we design a high-order well-balanced and positivity-preserving finite-difference AWENO scheme for solving the SWEs with or without dry areas. In this numerical framework, the hydrostatic reconstruction (HR) method is used with two main advantages: 1) The well-balanced property is achieved using the HR method with arbitrary monotone fluxes and the reformulated source terms. 2) The first-order scheme with the Lax-Friedrichs (LF) flux and the HR method maintains the positivity of the water height with an appropriate time step. Therefore, a simple positivity-preserving limiter conjugating the high-order reconstructed flux with the first-order positivity-preserving LF flux is introduced. One- and two-dimensional classical examples with or without dry areas demonstrate that the proposed scheme is high-order, well-balanced, and positivity-preserving.

Uncertainty quantification of turbulent systems via physically consistent and data-informed reduced-order models

This work presents a data-driven, energy-conserving closure method for the coarse-scale evolution of the mean and covariance of turbulent systems. Spatiotemporally non-local neural networks are employed for calculating the impact of non-Gaussian effects to the low-order statistics of dynamical systems with an energy-preserving quadratic nonlinearity. This property, which characterizes the advection term of turbulent flows, is encoded via an appropriate physical constraint in the training process of the data-informed closure. This condition is essential for the stability and accuracy of the simulations as it appropriately captures the energy transfers between unstable and stable modes of the system. The numerical scheme is implemented for a variety of turbulent systems, with prominent forward and inverse energy cascades. These problems include prototypical models such as an unstable triad system and the Lorentz-96 system, as well as more complex models: The two-layer quasi-geostrophic flows and incompressible, anisotropic jets where passive inertial tracers are being advected on. Training data are obtained through high-fidelity direct numerical simulations. In all cases, the hybrid scheme displays its ability to accurately capture the energy spectrum and high-order statistics of the systems under discussion. The generalizability properties of the trained closure models in all the test cases are explored, using out-of-sample realizations of the systems. The presented method is compared with existing first-order closure schemes, where only the mean equation is evolved. This comparison showcases that correctly evolving the covariance of the system outperforms first-order schemes in accuracy, at the expense of increased computational cost.

A Mass-Preserving Two-Step Lagrange–Galerkin Scheme for Convection-Diffusion Problems

A mass-preserving two-step Lagrange–Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of L^2-theory. The introduced scheme maintains the advantages of the Lagrange–Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-preserving property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-preserving scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the ell ^infty (L^2)- and ell ^2(H^1_0)-norms. For the time increment Delta t, the mesh size h and a conforming finite element space of polynomial degree k in {mathbb {N}}, the convergence order is of O(Delta t^2 + h^k) in the ell ^infty (L^2)cap ell ^2(H^1_0)-norm and of O(Delta t^2 + h^{k+1}) in the ell ^infty (L^2)-norm if the duality argument can be employed. Error estimates of O(Delta t^{3/2}+h^k) in discrete versions of the L^infty (H^1_0)- and H^1(L^2)-norm are additionally proved. Numerical results confirm the theoretical convergence orders in one, two and three dimensions.

Positivity-preserving entropy stable schemes for the 1-D compressible Navier-Stokes equations: High-order flux limiting

This is the second of two companion papers, in which we develop and analyze a new class of positivity-preserving, entropy stable spectral collocation schemes of arbitrary order of accuracy for the 1-D compressible Navier-Stokes equations. The key distinctive property of the proposed methodology is that it is proven to guarantee the pointwise positivity of thermodynamic variables for compressible viscous flows. The new schemes are constructed by combining a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable method developed in the companion paper [1]. This first-order scheme is discretized on the same Legendre-Gauss-Lobatto points used for the high-order counterpart. As a result, no interpolation or restriction is required between high- and low-order elements which are coupled by using the simultaneous approximation term (SAT) penalty method. To provide positivity preservation and excellent discontinuity-capturing properties, the Navier-Stokes equations are regularized by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The high- and low-order schemes are combined by using a limiting procedure, so that the resultant scheme provides conservation, guarantees pointwise positivity of thermodynamic variables, preserves the design–order of accuracy of the corresponding high-order baseline spectral collocation method for smooth solutions, and satisfies the discrete entropy inequality, thus facilitating a rigorous L2-stability proof for the symmetric form of the discretized Navier-Stokes equations. The proposed methodology is very general and can be extended to multiple dimensions and other summation-by-parts SAT-type schemes.

A New Fourth-Order Predictor–Corrector Numerical Scheme for Heat Transfer by Darcy–Forchheimer Flow of Micropolar Fluid with Homogeneous–Heterogeneous Reactions

This paper proposes a numerical scheme for solving linear and nonlinear differential equations obtained from the mathematical modeling of a flow phenomenon. The scheme is constructed on two grid points. It is a two-stage, or predictor–corrector type, scheme whose first stage (the predictor stage) comprises a forward Euler scheme. The stability region of the proposed scheme is larger than that of the first-order forward Euler scheme. A problem is constructed, comprised of a mathematical model for the Darcy–Forchheimer flow of micropolar fluid over a stretching sheet, and is modified using partial differential equations (PDEs) by incorporating the effects of homogeneous–heterogeneous reactions. A set of PDEs is further reduced into ordinary differential equations (ODEs) by several transformations and is solved using the proposed numerical scheme. By comparing the results obtained using the proposed scheme with those obtained using the existing forward Euler scheme, it can be observed that the proposed scheme achieved a smaller absolute error. The obtained results show that the angular velocity profile displayed dual behavior according to increases in the values of the microrotation and coupling constant parameters. As part of our research, we conducted a comparison with other existing schemes. The findings of this study can serve as a helpful guide for future investigations into fluid flow in closed-off industrial settings.

Vibro-acoustic analysis of a circumferentially coupled composite laminated annular plate backed by double cylindrical acoustic cavities

The current work focuses on the vibro-acoustic analysis of fluid-structure coupling system composed of a circumferentially coupled composite laminated annular plate and two cylindrical acoustic cavities considering arbitrary acoustic impedance. By employing the unified Ritz energy scheme in conjunction with the first-order shear lamination scheme, the acoustic and structural models in the studied physical system are reasonably established, where the required structural and plate-cavity continuity conditions are taken into account by means of the coupled spring technology and work, respectively. One-dimensional and two-dimensional orthogonal polynomials with the aid of the Fourier harmonic functions are separately applied to construct the variables functions of structural and acoustic domains without meshing requirements. The convergence and correctness of the developed prediction model including various physical parameters are carefully investigated through several comparative examples as well as vibro-acoustic experiment analysis, which illustrates the good insight in the vibro-acoustic mechanism of the studied physical system. Some parametric analysis respecting the effect of structural/acoustic boundaries, cavity depth, coupling radius and material parameter etc. on the coupled vibro-acoustic performance is implemented in detail, which may give some guidance for the low-noise pre-design of circumferentially coupled laminated annular plate.

Time integrators for dispersive equations in the long wave regime

We introduce a novel class of time integrators for dispersive equations which allow us to reproduce the dynamics of the solution from the classical ε = 1 \varepsilon = 1 up to long wave limit regime ε ≪ 1 \varepsilon \ll 1 on the natural time scale of the PDE t = O ( 1 ε ) t= \mathcal {O}(\frac {1}{\varepsilon }) . Most notably the global error of our new schemes is of order τ ε \tau \varepsilon (for the first-order scheme) and τ 2 ε \tau ^2 \varepsilon (for the second-order scheme) on time intervals of length O ( 1 ε ) \mathcal {O}\left ( \frac {1}{\varepsilon }\right ) .

Mixed-precision explicit stabilized Runge–Kutta methods for single- and multi-scale differential equations

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.