Second-order Accuracy Research Articles

Quadratic constraint consistency in the projection-free approximation of harmonic maps and bending isometries

We devise a projection-free iterative scheme for the approximation of harmonic maps that provides a second-order accuracy of the constraint violation and is unconditionally energy stable. A corresponding error estimate is valid under a mild but necessary discrete regularity condition. The method is based on the application of a BDF2 scheme and the considered problem serves as a model for partial differential equations with holonomic constraint. The performance of the method is illustrated via the computation of stationary harmonic maps and bending isometries.

A new linearized ADI compact difference method on graded meshes for a nonlinear 2D and 3D PIDE with a WSK

In this work, a new linearized alternating direction implicit (ADI) compact difference method (CDM) is proposed for solving nonlinear two-dimensional (2D) and three-dimensional (3D) partial integrodifferential equation (PIDE) with a weakly singular kernel (WSK). The time derivative is treated by Crank-Nicolson (CN) method and the Riemann-Liouville (R-L) integral by product integration (PI) rule on graded meshes. The linear interpolation combining with Taylor formula is applied in time to deal with nonlinear term v∇v in interval (t0,t1), and linear interpolation concerning two previous time points is employed to deal with v∇v in intervals (tn−1,tn),n≥2. A linearized semi-discrete scheme is obtained, which can achieve second-order convergence in time. Then via introducing two kinds of compact difference operators to discretize the spatial derivatives. To improve the computing efficiency, we construct a ADI compact difference method. It is the first time that the ADI compact difference method is applied for the nonlinear 2D and 3D PIDE with a WSK. The advantage of our proposed scheme is that it not only has second-order accuracy in time and fourth-order accuracy in space, but also fast computational speed, just by solving the linear coupled equations for tridiagonal matrices. In addition, we prove the existence, uniqueness and convergence of 2D scheme. Four numerical examples in 2D and 3D are present to demonstrate our proposed method.

A hybrid a posteriori MOOD limited lattice Boltzmann method to solve compressible fluid flows – LBMOOD

In this paper we blend two lattice-Boltzmann (LB) numerical schemes with an a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of conservation laws in 1D and 2D. The first LB scheme is robust to the presence of shock waves but lacks accuracy on smooth flows. The second one has a second-order of accuracy but develops non-physical oscillations when solving steep gradients. The MOOD paradigm produces a hybrid LB scheme via smooth and positivity detectors allowing to gather the best properties of the two LB methods within one scheme. Indeed, the resulting scheme presents second order of accuracy on smooth solutions, essentially non-oscillatory behavior on irregular ones, and, an ‘almost fail-safe’ property concerning positivity issues. The numerical results on a set of sanity test cases and demanding ones are presented assessing the appropriate behavior of the hybrid LBMOOD scheme in 1D and 2D.

A novel relaxed modified pipe hydrodynamic model for mixed flow simulation

In this study, we propose a novel modified pipe hydrodynamic model with a partial relaxation approach and develop a corresponding numerical algorithm within the framework of the finite volume method for its computation to enable robust, accurate, and comprehensive pipe modeling. The proposed numerical model provides some key improvements compared to existing numerical schemes for simulating mixed pipe flows: (1) it is capable of preserving both moving- and stationary-water steady states, and (2) it effectively reduces post-bore oscillations while maintaining second-order accuracy in the non-pressurized regime for unsteady pipe flows. Such improvements have been verified against theoretical and experimental data through tests involving both smooth and sharp-gradient steady and unsteady flows across various regimes in straight and locally bent circular pipes, which are fundamental elements of stormwater drainage, sewage networks, and industrial process pipelines. The proposed pipe flow model can be further improved regarding post-pore stability by utilizing specific post-bore oscillation suppression techniques, and it can be integrated into a practical hydraulic modeling tool for pipeline networks in future work.

A turbulence model: Second-order temporal accuracy

This paper develops the defect-deferred correction method of finite element discretization to solve a turbulence model. The method consists of two phases: (1) implementing an artificial viscosity term as a defect step and (2) applying a correction strategy. This method can not only achieve second-order accuracy in time but also simulate small viscosity problem. Moreover, the stability and convergence of the defect step and the deferred correction step are proven, respectively. Finally, we perform several numerical examples to validate the theoretical analysis and illustrate the effectiveness of the proposed method.

Numerical Solution of Burgers Equation Using Finite Difference Methods: Analysis of Shock Waves in Aircraft Dynamics

In this research, the Lax, the Upwind, and the MacCormack finite difference methods are applied to the experimental solving of the one-dimensional (1D) unsteady Burger's Equation, a Hyperbolic Partial Differential Equation. These three numerical analysis-solving methods are implemented for accurate modeling of shock wave behavior high-speed flows that are necessary for aerospace engineering design. This research analysis proves that the MacCormack technique is the one that treats the differential equations with second-order accuracy. This method is quite preferred when it comes to numerical simulations because of its advanced level of accuracy. Although the Upwind and Lax methods are slightly less accurate, they show the development of shock waves that give visualizations to better understand the flow dynamics. Also, in this study, the impact of varying viscosity coefficients on fluid flow characteristics by using the lax (a numerical method for solving the viscous Burgers equation) is investigated. This identification of the phenomenon sheds light on the behavior of boundary layers, which, in turn, can be used to improve the design of high-speed vehicles and lead to a greater understanding of the area of ​​fluid dynamics.

Second order conservative Lagrangian DG schemes for compressible flow and their application in preserving spherical symmetry in two-dimensional cylindrical geometry

In this paper, we construct a class of second-order cell-centered Lagrangian discontinuous Galerkin (DG) schemes for the two-dimensional compressible Euler equations on quadrilateral meshes. This Lagrangian DG scheme is based on the physical coordinates rather than the fixed reference coordinates, hence it does not require studying the evolution of the Jacobian matrix for the flow mapping between the different coordinates. The conserved variables are solved directly, and the scheme can preserve the conservation property for mass, momentum and total energy. The strong stability preserving (SSP) Runge-Kutta (RK) method is used for the time discretization. Furthermore, there are two main contributions. Firstly, differently from the previous work, we design a new Lagrangian DG scheme which is truly second-order accurate for all the variables such as density, momentum, total energy, pressure and velocity, while the similar DG schemes in the literature may lose second-order accuracy for certain variables, as shown in numerical experiments. Secondly, as an extension and application, we develop a particular Lagrangian DG scheme in the cylindrical geometry, which is designed to be able to preserve one-dimensional spherical symmetry for all the linear polynomials in two-dimensional cylindrical coordinates when computed on an equal-angle-zoned initial grid. The distinguished feature is that it can maintain both the spherical symmetry and conservation properties, which is very important for many applications such as implosion problems. A series of numerical experiments in the two-dimensional Cartesian and cylindrical coordinates are given to demonstrate the good performance of the Lagrangian DG schemes in terms of accuracy, symmetry and non-oscillation.

Construction of Second-Order Finite Difference Schemes for Diffusion-Convection Problems of Multifractional Suspensions in Coastal Marine Systems

Introduction. This paper addresses an initial-boundary value problem for the transport of multifractional suspensions applied to coastal marine systems. This problem describes the processes of transport, deposition of suspension particles, and the transitions between its various fractions. To obtain monotonic finite difference schemes for diffusion-convection problems of suspensions, it is advisable to use schemes that satisfy the maximum principle. When constructing a finite difference scheme that adheres to the maximum principle, it is desirable to achieve second-order spatial accuracy for bothinterior and boundary points of the domain under study. Materials and Methods. This problem presents certain difficulties when considering the boundaries of the geometric domain, where boundary conditions of the second and third kinds are applied. In these cases, to maintain second-order approximation accuracy, an “extended” grid is introduced (a grid supplemented with fictitious nodes). The guidelineis the approximation of the given boundary conditions using the central difference formula, with the exclusion of the concentration function at the fictitious node from the resulting expressions. Results. Second-order accurate finite difference schemes for the diffusion-convection problem of multifractional suspensions in coastal marine systems are constructed. Discussion and Conclusion. The proposed schemes are not absolutely stable, and a detailed analysis of stability and convergence, particularly concerning the grid step ratio, remains an important problem that the author plans to address in the future.

Mathematical Modelling of Spatially Inhomogeneous Non-Stationary Interaction of Pests with Transgenic and Non-Modified Crops Considering Taxis

Introduction. This paper addresses a unified spatially inhomogeneous, non-stationary model of interaction between genetically modified crop resources (corn) and the corn borer pest, which is also present on a relatively small section of non-modified corn. The model assumes that insect pests influence both types of crops and are capable of independent movement (taxis) towards the gradient of plant resources. It also considers diffusion processes in the dynamics of all components of the unified model, biomass growth, genetic characteristics of both types of plant resources, processes of crop consumption, phenomena of growth and degradation, diffusion, and mutation of pests. The model allows for predictive calculations aimed at reducing crop losses and increasing the resistance of transgenic crops to pests by slowing down the natural mutation rate of the pest. Materials and Methods. The mathematical model is an extension of Kostitsin’s model and is formulated as an initial-boundary value problem for a nonlinear system of convection-diffusion equations. These equations describe the spatiotemporal dynamics of biomass density changes in two types of crops — transgenic and non-modified — as well as the specific populations (densities) of three genotypes of pests (the corn borer) resulting from mutations. The authors linearized the convection-diffusion equations by applying a time-lag method on the time grid, with nonlinear terms from eachequation taken from the previous time layer. The terms describing taxis are presented in a symmetric form, ensuring the skew-symmetry of the corresponding continuous operator and, in the case of spatial grid approximation, the finite-difference operator. Results. A stable monotonic finite-difference scheme is developed, approximating the original problem with second-order accuracy on a uniform 2D spatial grid. Numerical solutions of model problems are provided, qualitatively corresponding to observed processes. Solutions are obtained for various ratios of modified and non-modified sections of the field. Discussion and Conclusion. The obtained results regarding pest behavior, depending on the type of taxis, could significantly extend the time for pests to acquire Bt resistance. The concentration dynamics of pests moving in the direction of the food gradient differs markedly from the concentration of pests moving towards a mate for reproduction.

High-Accuracy Positivity-Preserving Finite Difference Approximations of the Chemotaxis Model for Tumor Invasion.

Numerical simulation of the complex evolution process for tumor invasion plays an extremely important role in-depth exploring the bio-taxis phenomena of tumor growth and metastasis. In view of the fact that low-accuracy numerical methods often have large errors and low resolution, very refined grids have to be used if we want to get high-resolution simulating results, which leads to a great deal of computational cost. In this paper, we are committed to developing a class of high-accuracy positivity-preserving finite difference methods to solve the chemotaxis model for tumor invasion. First, two unconditionally stable implicit compact difference schemes for solving the model are proposed; second, the local truncation errors of the new schemes are analyzed, which show that they have second-order accuracy in time and fourth-order accuracy in space; third, based on the proposed schemes, the high-accuracy numerical integration idea of binary functions is employed to structure a linear compact weighting formula that guarantees fourth-order accuracy and nonnegative, and then a positivity-preserving and time-marching algorithm is established; and finally, the accuracy, stability, and positivity-preserving of the proposed methods are verified by several numerical experiments, and the evolution phenomena of tumor invasion over time are numerically simulated and analyzed.

Influence of electron cyclotron current drive on the pressure crashes caused by the 2/1 double tearing modes

Abstract A module with self-consistent evolution of driven current is developed and coupled with the resistive-MHD equations in the three-dimensional, toroidal, and nonlinear simulation code (CLT). The driven current equation is solved with a second-order accuracy symmetric scheme, which exhibits good conservation properties. With the new module, we find that the driven current can self-consistently concentrate inside the magnetic island when the parallel diffusion of the driven current is sufficiently large. The efficiency of the driven current on tearing mode suppression will then be much higher than those with stationary distributions. With the new module, the influence of electron cyclotron current drive (ECCD) on the nonlinear evolution of the 2/1 double tearing modes (DTMs) is investigated. When co-ECCD deposits on the outer resonant surface, the local magnetic shear is reduced, and the growth rates of the DTMs decrease; if ctr-ECCD deposits on the outer resonant surface, the local magnetic shear increases, and the DTMs become more unstable. However, things will be different if ECCD deposits on the inner resonant surface since the local magnetic shear is negative. The co-ECCD deposited on the inner resonant surface increases the negative shear and then promotes the growth of the DTMs; while the ctr-ECCD suppresses the DTMs. It is also found that the off-axis and central pressure crashes associated with the 2/1 DTMs can be converted to each other by properly depositing the driven current. To convert a central crash to an off-axis crash, the co-ECCD should be deposited on the outer resonant surface, or the ctr-ECCD deposited on the inner resonant surface. While, the co-ECCD should be deposited on the inner rational surface, or the ctr-ECCD deposited on the outer rational surface to convert an off-axis crash to a central crash. The co- or ctr-ECCD should be larger than a threshold for such transitions, and the threshold value is mainly determined by the location of the inner resonant surface.

Numerical solution of the mixed boundary value problem for the heat equation in two-dimensional domains of complex shape

A finite-difference computational algorithm is proposed for solving a mixed boundary value problem for heat equation given in a two-dimensional domains of complex shape. To solve the problem, generalized curvilinear coordinates are used. The physical domain is mapped to the computational domain (unit square) in the space of generalized coordinates. The original problem is written in curvilinear coordinates and approximated on a uniform grid in the computational domain. The obtained results are mapped on a non-uniform boundary-fitted difference grid in the physical domain. The second-order approximations of mixed Neumann – Dirichlet boundary conditions are constructed. The results of computational experiments are presented. The second order of accuracy of the presented computational algorithm is confirmed.

Computing Euclidean distance and maximum likelihood retraction maps for constrained optimization

Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence retraction maps are used to approximate the exponential map and return to the manifold. For many common matrix manifolds, retraction maps are available, with more or less explicit formulas. For implicitly-defined manifolds, suitable retraction maps are difficult to compute. We therefore develop an algorithm which uses homotopy continuation to compute the Euclidean distance retraction for any implicitly-defined submanifold of Rn, and prove convergence results.We also consider statistical models as Riemannian submanifolds of the probability simplex with the Fisher metric. Replacing Euclidean distance with maximum likelihood results in a map which we prove is a retraction. In fact, we prove the retraction is second-order; with the Levi-Civita connection associated to the Fisher metric, it approximates geodesics to second-order accuracy.

A coupled immersed boundary method and wall modelling framework for high-Reynolds number flows over complex terrain

We present the development and validation of an immersed boundary method (IBM) along with a wall model (WM) to simulate atmospheric boundary-layer flows over complex terrain. The framework presented here has two novel aspects over standard IBM implementations. First, the underlying schemes are global in nature and require specification of values throughout the solid region. Second, to enable high-Reynolds number simulations, a wall model is coupled to the IBM. The proposed numerical framework is shown to have second-order accuracy. The framework is validated by simulating flow over a 2D as well as 3D cosine-squared hill and comparing to previously published experimental results. The mean velocity and turbulence intensity are reproduced accurately by our LES. The flow over the Bolund hill, marked by steep slopes which makes this a challenging test case, is also simulated and results are compared to field observations showing good agreement. The framework also accurately reproduces the turbulent statistics in the wake of a turbine situated on a flat terrain, with the flat terrain modelled using the IBWM framework, thus demonstrating its applicability to high-Reynolds number atmospheric and wind farm flows over complex terrain.

A multiscale stochastic particle method based on the Fokker-Planck model for nonequilibrium gas flows

The Fokker-Planck (FP) model characterizes the intermolecular collisions as continuous stochastic processes in velocity space. Compared to the Direct Simulation Monte Carlo (DSMC) method, which is grounded in the Boltzmann equation, the stochastic particle method based on the FP model, referred to as the SP-FP method, demonstrates a relative improvement in computational efficiency. However, the traditional SP-FP method, utilizing operator splitting scheme, encounters analogous limitations to DSMC, thereby limiting the application of large temporal-spatial discretization for simulating near-continuum and continuum flows. In this study, two critical advancements are introduced to improve the traditional SP-FP method. First, the exact integration scheme of the ellipsoidal-statistical FP (ESFP) model is derived, enabling the replication of the theoretical solutions for homogeneous relaxation with any time step. Furthermore, leveraging the integration scheme of the ESFP model, we propose a Multiscale Stochastic Particle (MSP) method. The MSP method quantifies the numerical errors stemming from the operator splitting scheme and corrects them in the relaxation step. This capability allows the MSP method to be employed with larger temporal-spatial discretization for inhomogeneous problems, achieving second-order accuracy while retaining implementation simplicity. Application of the MSP method to various benchmark problems, including Couette flow, Fourier flow, Poiseuille flow, Sod tube flow, Lid-driven cavity flow, and flow through a slit, demonstrates its promise for simulating multiscale nonequilibrium gas flows with high computational efficiency and accuracy.

An unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn–Hilliard equation on generalized SAV approach

Abstract An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn–Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., A1: $0<r_{k}:=\tau _{k}/\tau _{k-1}< r_{\max }\approx 4.8645$, we establish a rigorous error estimate in $H^{1}$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that the presented adaptive time-step scheme only requires solving one linear system with constant coefficients at each time step. In our analysis, the first-consistent BDF1 for the first step does not bring the order reduction in $H^{1}$-norm. The $H^{1}$ bound of numerical solution under periodic boundary conditions can be derived without any restriction (such as zero mean of the initial data). Finally, numerical examples are provided to verify our theoretical analysis and the algorithm efficiency.

A new variational integrator for constrained mechanical system dynamics

A new variational integrator is proposed to solve constrained mechanical systems. The main distinguishing feature of the present integrator comes from the distinct discretization of Lagrangians based on the Hamilton's principle in its most general form. Specifically, Hermite interpolation is used for discrete positions, which provides at least C1 continuity for generalized coordinates. The velocities and momentums are interpolated using quadratic polynomials for the consistency, such that the kinematic relation between velocities and positions can be exactly satisfied. Meanwhile, the Gauss-Legendre quadrature rule is employed to guarantee the accuracy of discrete Lagrange equations. To tackle constrained mechanical systems, a coordinate partition approach is used to eliminate the constraint equations. The local incremental rotation vector is exploited to get rid of rotation singularities in spatial problems. Moreover, an adaptive stepsize strategy is implemented to improve the efficiency. The strengths of the new integrator lie in the accessible large step sizes in the simulation and its global second-order accuracy for positions as well as velocities. Several examples are performed and analyzed to validate its accuracy and capabilities.

Analysis of efficient preconditioner for solving Poisson equation with Dirichlet boundary condition in irregular three-dimensional domains

This paper analyzes modified ILU (MILU)-type preconditioners for efficiently solving the Poisson equation with Dirichlet boundary conditions on irregular domains. In [1], the second-order accuracy of a finite difference scheme developed by Gibou et al. [2] and the effect of the MILU preconditioner were presented for two-dimensional problems. However, the analyses do not directly extend to three-dimensional problems. In this paper, we first demonstrate that the Gibou method attains second-order convergence for three-dimensional irregular domains, yet the discretized Laplacian exhibits a condition number of O(h−2) for a grid size h. We show that the MILU preconditioner reduces the order of the condition number to O(h−1) in three dimensions. Furthermore, we propose a novel sectored-MILU preconditioner, defined by a sectorized lexicographic ordering along each axis of the domain. We demonstrate that this preconditioner reaches a condition number of order O(h−1) as well. Sectored-MILU not only achieves a similar or better condition number than conventional MILU but also improves parallel computing efficiency, enabling very efficient calculation when the dimensionality or problem size increases significantly. Our findings extend the feasibility of solving large-scale problems across a range of scientific and engineering disciplines.

Soliton solutions of derivative nonlinear Schrödinger equations: Conservative schemes and numerical simulation

In this paper, we numerically study soliton solutions of derivative nonlinear Schrödinger equations based on several conservative finite difference methods. All schemes own second-order accuracy with the convergence order O(τ2+h2) in the discrete L∞-norm, where h denotes the spatial step size and τ denotes the temporal step size. We show that difference schemes preserve some discrete counterparts of continuous conservation laws, and all these schemes are solvable. Extensive numerical examples with soliton solutions are carried out to verify the theoretical results. These results manifest that our schemes have potential application to soliton propagation in optical fibers.

Second-Order Modified Nonstandard Explicit Euler and Explicit Runge–Kutta Methods for n-Dimensional Autonomous Differential Equations

Nonstandard finite-difference (NSFD) methods, pioneered by R. E. Mickens, offer accurate and efficient solutions to various differential equation models in science and engineering. NSFD methods avoid numerical instabilities for large time steps, while numerically preserving important properties of exact solutions. However, most NSFD methods are only first-order accurate. This paper introduces two new classes of explicit second-order modified NSFD methods for solving n-dimensional autonomous dynamical systems. These explicit methods extend previous work by incorporating novel denominator functions to ensure both elementary stability and second-order accuracy. This paper also provides a detailed mathematical analysis and validates the methods through numerical simulations on various biological systems.