Second-order Accuracy In Time Research Articles

Pulse supercharging phenomena in a water-filled pipe and a universal prediction model of optimal pulse frequency

A water hammer is an important natural phenomenon that can be used to fracture rock with enhanced local water pressure. The oscillatory injection of a column of water can be used to make a pipe water hammer. However, the optimal injection frequency to create a water hammer has not yet been found. The main reason for this is that the distribution of fluid pressure and its variation are unclear inside a pipe. In this study, we demonstrate for the first time that there can be significant supercharging phenomena and a law governing their appearance in a water-filled pipe. We first find the optimal pulse frequency to reproduce the supercharging process. We also clarify the supercharging mechanism at an optimal frequency. First, a simplified pipe model is adopted, and weakly compressible Navier–Stokes equations are developed to simulate the flow of water in pulse hydraulic fracturing (PHF). The computation code is developed using the MacCormack method, which has second-order accuracy in time and space. The computation codes and program are validated using experimental data of weakly compressible flows. Then, the square pulse effects are studied inside a pipe, including the effects of pulse frequency, amplitude, pipe length, diameter, and wave speed. Finally, a new universal frequency model is built to describe the relationship among optimal pulse frequency, wave speed, and pipe length. The results show that in square PHF, there is a family of frequencies for which the fluid peak pressure can be significantly enhanced, and these frequencies include the optimal pulse frequency. The optimal frequency of a square pulse depends on the pipe length and wave speed. At the optimal pulse frequency, the maximum peak pressure of the fluid can be increased by 100% or more, and cavitation occurs. These new landmark findings are very valuable for understanding pulse supercharging in an internal water wave. In addition, a new universal frequency model is built to predict optimal pulse frequency. This study identifies an evolution law of peak pressure inside a pipe and proposes a practical frequency-control model for the first time, which can provide a theoretical guide for PHF design.

High-Performance Uniform Damping Model for Response History Analysis in OpenSees

ABSTRACT A realistic and practical damping model is essential for structural dynamics. In this study, the second-order accurate time discretization of a high-performance uniform damping model for response history analysis is derived. Its stability and accuracy are proven, and its object-oriented implementation is introduced in OpenSees. The proposed time discretization accurately represents energy dissipation in individual components with minimal additional computational costs. It applies to various time integrators, element formulations, and structural systems, allowing for high-fidelity evaluation of structural performance and ensuring public safety. These advantages are illustrated through detailed case studies.

An immersed boundary/multi-relaxation time lattice Boltzmann method on adaptive octree grids for the particle-resolved simulation of particle-laden flows

We present an immersed boundary/multi-relaxation time lattice Boltzmann method for the particle-resolved simulation of particle-laden flows. The no-slip boundary condition is handled by an explicit feedback immersed boundary method for fixed and moving particles with arbitrary shape. Two special treatments are applied to improve the simulation stability and accuracy: (i) a smoothed discrete delta function (Yang et al. (2009) [47]) is adopted to suppress the spurious force oscillations and (ii) a multi-relaxation time collision operator is utilized to fix the viscosity-dependent numerical slip error of the immersed boundaries. To further reduce the computational effort, we extend the method from fixed and uniform Cartesian grids to adaptive quadtree/octree grids by implementing it in the open-source software Basilisk. A Lax-Wendroff streaming operator is adopted in our method to retain second-order accuracy in both space and time on non-uniform grids, as well as to maintain a unified time scale over the entire computational domain. Consequently, one collision-streaming operation only is required per time step at all grid levels. We test our proposed method on a set of validation cases corresponding to assorted incompressible flows with fixed and moving rigid particles in both 2D and 3D. The accuracy and robustness of our method are demonstrated via thorough comparison with the analytical, experimental and numerical data provided in the literature.

A truly self-starting composite isochronous integration analysis framework for first/second-order transient systems

Due to the lack of and limited work done for first/second-order systems useful for multidisciplinary problems, this paper proposes a new and novel composite isochronous integration ([i-Integration]) analysis framework to solve first/second-order transient systems via a single computational framework. The main contributions are summarized as follows: (1) The ρ∞-Bathe method is newly reformed to an identical truly self-starting representation with the absence of acceleration and is more efficient for practical applications; (2) The novel [i-Integration] process is applied to automatically generate a family of time integration algorithms for first/second-order transient systems, whilst preserving features of second-order time accuracy, unconditional stability, controllable numerical dissipation in high-frequency, and truly self-starting; (3) The newly proposed algorithms design framework provides a unified toolkit to solve first/second-order transient systems in a single analysis and hence is more effective for coupled problems and numerical analysis of fluid/heat transfer/structure-type transient simulations. Numerical examples encompassing a manufactured example, the C-F heat conduction model, and the thermal stress wave problem are demonstrated to validate the proposed algorithms.

A novel time‐domain numerical methodology for the electromagnetic analysis of an H‐plane tee power divider

A time-domain numerical methodology for the transient analysis of multisection waveguide power dividers is proposed. Waveguide power dividers are essential components of microwave systems for high-power applications. A numerical strategy based on the 3D discontinuous Galerkin time-domain finite element method is used to estimate the electromagnetic propagation inside a power divider with three ports. The proposed approach leads to simulations that are fast, robust, and accurate, this being the result of a parallel local discretisation of the full-wave time-domain Maxwell's equations on unstructured tetrahedral meshes. The stability of the overall numerical scheme is obtained using an upwind numerical flux, an implicit second-order accurate time integration scheme, and Whitney's edge elements. Good agreements are obtained when the computed results are compared with other HFSS simulation results, and with measurement results found in the literature.

A Unified Lattice Boltzmann Model for Fourth Order Partial Differential Equations with Variable Coefficients.

In this work, a unified lattice Boltzmann model is proposed for the fourth order partial differential equation with time-dependent variable coefficients, which has the form . A compensation function is added to the evolution equation to recover the macroscopic equation. Applying Chapman-Enskog expansion and the Taylor expansion method, we recover the macroscopic equation correctly. Through analyzing the error, our model reaches second-order accuracy in time. A series of constant-coefficient and variable-coefficient partial differential equations are successfully simulated, which tests the effectiveness and stability of the present model.

Fast Compact Difference Scheme for Solving the Two-Dimensional Time-Fractional Cattaneo Equation

The time-fractional Cattaneo equation is an equation where the fractional order α∈(1,2) has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an equation in two space dimensions. First, we obtain the space’s semi-discrete numerical scheme by using the compact difference operator in the spatial direction. Then, the semi-discrete scheme is converted to a low-order system by means of order reduction, and the fully discrete compact difference scheme is presented by applying the L2-1σ formula. To improve the computational efficiency, we adopt the fast discrete Sine transform and sum-of-exponentials techniques for the compact difference operator and L2-1σ difference operator, respectively, and derive the improved scheme with fast computations in both time and space. That aside, we also consider the graded meshes in the time direction to efficiently handle the weak singularity of the solution at the initial time. The stability and convergence of the numerical scheme under the uniform meshes are rigorously proven, and it is shown that the scheme has second-order and fourth-order accuracy in time and in space, respectively. Finally, numerical examples with high-dimensional problems are demonstrated to verify the accuracy and computational efficiency of the derived scheme.

The Construction of High-Order Robust Theta Methods with Applications in Subdiffusion Models

An exponential-type function was discovered to transform known difference formulas by involving a shifted parameter θ to approximate fractional calculus operators. In contrast to the known θ methods obtained by polynomial-type transformations, our exponential-type θ methods take the advantage of the fact that they have no restrictions in theory on the range of θ such that the resultant scheme is asymptotically stable. As an application to investigate the subdiffusion problem, the second-order fractional backward difference formula is transformed, and correction terms are designed to maintain the optimal second-order accuracy in time. The obtained exponential-type scheme is robust in that it is accurate even for very small α and can naturally resolve the initial singularity provided θ=−12, both of which are demonstrated rigorously. All theoretical results are confirmed by extensive numerical tests.

The Crank–Nicolson mixed finite element method for the improved system of time-domain Maxwell’s equations

In this paper, we mainly establish the Crank–Nicolson (CN) mixed finite element (MFE) method for the system of time-domain Maxwell’s equations with a lossy medium. To this end, we first deduce the two-dimensional (2D) improved system of time-domain Maxwell’s equations with the lossy medium, construct the time semi-discretized CN format for the 2D improved system of Maxwell’s equations, and discuss the existence, stability, and error estimates of time semi-discrete solutions. And then, we construct a fully discretized CN MFE (FDCNMFE) format with both symmetry and positive definiteness, and analyze the existence, stability, and errors of FDCNMFE solutions. Furthermore, we make use of some numerical tests to verify the correctness of theoretical results in the 2D case. Finally, as a generalization, we also give the corresponding methods and results for three-dimensional (3D) improved system of time-domain Maxwell’s equations in the appendix A. This paper has advantages in at least the following four aspects. First, the study process herein is to discrete time and then discrete space variables so as to save the semi-discretized MFE discussion for spatial variables, which is a new attempt. Second, the magnetic field strength in the improve system of Maxwell’s equations is independent of the electric intensity such that their MFE subspaces can avoid the constraint of the Babuška-Brezzi (B-B) condition and can be freely chosen, so that the FDCNMFE solutions reach optimal order error estimates. Third, the 2D and 3D FDCNMFE formats reach the second-order accuracy in time and have both symmetry and positive definiteness so as to be solved easily. Fourth, both time semi-discretized CN solutions and the FDCNMFE solutions are unconditionally stable and unconditionally convergent.

Fully-discrete Spectral-Galerkin numerical scheme with second-order time accuracy and unconditional energy stability for the anisotropic Cahn–Hilliard Model

In this work, we construct a fully-discrete Spectral-Galerkin scheme for the anisotropic Cahn–Hilliard model. The scheme is based on the combination of a novel so-called explicit-Invariant Energy Quadratization method for time discretization and the Spectral-Galerkin approach for spatial discretization. The designed scheme only needs to solve several independent linear equations with constant coefficients at each time step, which demonstrates the high computational efficiency. The introduction of two auxiliary variables and the design of their associated auxiliary ODEs play a vital role in obtaining the linear structure and unconditional energy stability and thus avoiding the computation of a variable-coefficient system. The unconditional energy stability of the scheme is further rigorously proved, and the implementation process is given in detail. Through several 2D and 3D numerical simulations, we further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.

An implicit conformation tensor decoupling approach for viscoelastic flow simulation within the monolithic projection framework

The highly nonlinear nature of the system governing equations makes it difficult to simulate viscoelastic flows efficiently. In this paper, an implicit decoupling approach is proposed for the viscoelastic flow simulation with a monolithic projection method. The decoupling approach can be derived from the approximate block factorization at the matrix level, which has been successfully applied into the Newtonian flow simulations. In our work, we extend this approach to decouple the pressure, conformation tensor and velocity from the viscoelastic flow system sequentially. Firstly, the pressure-conformation tensor-velocity decoupling is realized by a two-step approximate factorization. By combining the conformation tensor and velocity together at the first-step factorization, the original efficient pressure Poisson solver for the Newtonian flow simulation can be directly adopted in the present framework for non-Newtonian flow simulation. Secondly, the conformation tensor components are further decoupled from each other by the present component-decoupling approach. Then all quantities, including pressure, conformation tensor and velocity, can be resolved without iteration. Additionally, the accuracy corrector is proposed to access the prior scheme limiter-estimation problem introduced by the implicit high-resolution scheme. We numerically demonstrate that all quantities preserve the second-order accuracy in time and space as expected by the theoretical splitting errors. Finally, several different types of canonical benchmark flow examples are conducted to validate the present solver, including laminar flow, turbulent drag-reducing flow and the rotation of a spherical particle immersed in a sheared viscoelastic fluid.

Numerical approximations of flow coupled binary phase field crystal system: Fully discrete finite element scheme with second-order temporal accuracy and decoupling structure

In this article, we first establish a new flow-coupled binary phase-field crystal model and prove its energy law. Then by using some newly introduced variables, we reformulate this three-phase model into an equivalent form, which makes it possible to construct a fully discrete linearized decoupling scheme with unconditional energy stability and second-order time accuracy to solve this model for the first time. The energy law of the reformulated model is also proved. Then we incorporate the explicit-IEQ (invariant energy quadratization) method for the nonlinear potentials, the projection method for the Navier-Stokes equations, the Crank-Nicolson method for time marching, and the finite element method for spatial discretization together to develop the fully discrete scheme for the reformulated and equivalent system. By using the nonlocal splitting technique, at each time step, only a few decoupled constant-coefficient elliptic equations are required to be solved, even though the original and reformulated models are much more complicated in the form. The developed algorithm is further proved to be unconditionally energy stable, and a detailed implementation process is also provided. Various numerical experiments in 2D and 3D are carried out to verify the effectiveness of the developed scheme, including the binary crystal growth under the action of shear flow and the sedimentation process of many binary particles.

Two Time-Stepping Schemes for Sub-Diffusion Equations with Singular Source Terms

Singular source terms in sub-diffusion equations may lead to the unboundedness of solutions, which will bring a severe reduction of convergence order of existing time-stepping schemes. In this work, we propose two efficient time-stepping schemes for solving sub-diffusion equations with a class of source terms mildly singular in time. One discretization is based on the Gr{\"u}nwald-Letnikov and backward Euler methods. First-order error estimate with respect to time is rigorously established for singular source terms and nonsmooth initial data. The other scheme derived from the second-order backward differentiation formula (BDF) is proved to possess second-order accuracy in time. Further, piecewise linear finite element and lumped mass finite element discretizations in space are applied and analyzed rigorously. Numerical investigations confirm our theoretical results.

A symmetric low-regularity integrator for nonlinear Klein-Gordon equation

In this work, we propose a symmetric exponential-type low- regularity integrator for solving the nonlinear Klein-Gordon equation under rough data. The scheme is explicit in the physical space, and it is efficient under the Fourier pseudospectral discretization. Moreover, it achieves the second-order accuracy in time without loss of regularity of the solution, and its time-reversal symmetry ensures the good long-time behavior. Error estimates are done for both semi- and full discretizations. Numerical results confirm the theoretical results, and comparisons illustrate the improvement of the proposed scheme over the existing methods.

The Dimensionality Reduction of Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors for the Unsteady Stokes Equation

By means of a proper orthogonal decomposition (POD) to cut down the dimensionality of unknown finite element (FE) solution coefficient vectors in the Crank–Nicolson (CN) mixed FE (CNMFE) method for two-dimensional (2D) unsteady Stokes equations in regard to vorticity stream functions, a reduced dimension recursive-CNMFE (RDR-CNMFE) method is constructed. In this case, the RDR-CNMFE method has the same FE basis functions and accuracy as the CNMFE method. The existence, stability, and errors of RDR-CNMFE solutions are analyzed by matrix analyzing, resulting in very simple theory analysis. Some numerical tries are used to check on the validity of the RDR-CNMFE method. The RDR-CNMFE method has second-order time accuracy and few unknowns so as to be able to shorten CPU runtime and retard the error cumulation in simulation calculating process, and improve real-time calculating accuracy.

Digital Simulations for Three-dimensional Nonlinear Advection-diffusion Equations Using Quasi-variable Meshes High-resolution Implicit Compact Scheme

A two-level implicit compact formulation with quasi-variable meshes is reported for solving three-dimensions second-order nonlinear parabolic partial differential equations. The new nineteen-point compact scheme exhibit fourth and second-order accuracy in space and time on a variable mesh steps and uniformly spaced mesh points. We have also developed an operator-splitting technique to implement the alternating direction implicit (ADI) scheme for computing the 3D advection-diffusion equation. Thomas algorithm computes each tri-diagonal matrix that arises from ADI steps in minimal computing time. The operator-splitting form is unconditionally stable. The improved accuracy is achieved at a lower cost of computation and storage because the spatial mesh parameters tune the mesh location according to solution values' behavior. The new method is successfully applied to the Navier-Stokes equation, advection-diffusion equation, and Burger's equation for the computational illustrations that corroborate the order, accuracies, and robustness of the new high-order implicit compact scheme. The main highlight of the present work lies in obtaining a fourth-order scheme on a quasi-variable mesh network, and its superiority over the comparable uniform meshes high-order compact scheme.

Digital Simulations for Three-dimensional Nonlinear Advection-diffusion Equations Using Quasi-variable Meshes High-resolution Implicit Compact Scheme

A two-level implicit compact formulation with quasi-variable meshes is reported for solving three-dimensions second-order nonlinear parabolic partial differential equations. The new nineteen-point compact scheme exhibit fourth and second-order accuracy in space and time on a variable mesh steps and uniformly spaced mesh points. We have also developed an operator-splitting technique to implement the alternating direction implicit (ADI) scheme for computing the 3D advection-diffusion equation. Thomas algorithm computes each tri-diagonal matrix that arises from ADI steps in minimal computing time. The operator-splitting form is unconditionally stable. The improved accuracy is achieved at a lower cost of computation and storage because the spatial mesh parameters tune the mesh location according to solution values' behavior. The new method is successfully applied to the Navier-Stokes equation, advection-diffusion equation, and Burger's equation for the computational illustrations that corroborate the order, accuracies, and robustness of the new high-order implicit compact scheme. The main highlight of the present work lies in obtaining a fourth-order scheme on a quasi-variable mesh network, and its superiority over the comparable uniform meshes high-order compact scheme.

Crank–Nicolson extrapolation and finite element method for the Oldroyd fluid with the midpoint rule

An efficient method is examined for the Oldroyd fluid which belongs to nonlinear integro-differential equations. This approach is approximately discretized by the FEM (i.e. finite element method) in space level, and a second-order Crank–Nicolson extrapolation in time. It reduces nonlinear equations to linear equations. Thus can greatly increase the computational efficiency. In order to approximate the integral term, we apply the “midpoint rule” approach. The error of the integral term to the midpoint rule is determined. We establish that the method is unconditionally stable and convergent. The L2-optimal error estimate with second order accuracy in time and space level is obtained without imposing any restriction on time-step size, while all previous works of higher-order scheme require certain time-step conditions. Moreover, corresponding numerical experiments are shown to demonstrate the accuracy, the unconditional stability and convergence of this method for the Oldroyd fluid. The structure is also suitable for the Navier–Stokes equations (i.e. ρ=0 in the Oldroyd fluid).

Error analysis of the second-order BDF finite element scheme for the thermally coupled incompressible magnetohydrodynamic system

In this paper, we consider the initial-boundary value problem for the three-dimensional incompressible magnetohydrodynamic coupled heat equation through the Boussinesq approximation. A finite element fully discrete scheme is proposed for approximating and solving this coupled system numerically, where the second-order extrapolation scheme is used for the discretization of time derivative terms and the mixed finite element method is used for the spatial discretization. Furthermore, we use the mini finite element to approximate the velocity and pressure, and use piecewise linear finite elements to approximate the magnetic field and temperature. We prove the unconditional stability of fully discrete scheme and derive the optimal second-order convergent accuracy in both time and spatial directions. Finally, numerical results are presented to illustrate the second-order convergence rates by taking the CFL condition of order 1.

A Numerical Scheme for Fractional Mixed Convection Flow Over Flat and Oscillatory Plates

Abstract A fractional scheme is proposed to solve time-fractional partial differential equations. According to the considered fractional Taylor series, the scheme is compact in space and provides fourth-order accuracy in space and second-order accuracy in fractional time. The scheme is conditionally stable when applied to the scalar fractional parabolic equation. The convergence of the scheme is demonstrated for the system of fractional parabolic equations. Moreover, a fractional model for heat and mass transfer of mixed convection flow over the flat and oscillatory plate is given. The radiation effects and chemical reactions are also considered. The scheme is tested on this model and the nonlinear fractional Burgers equation. It is found that it is more accurate than considering existing schemes in most of the regions of the solution domain. The compact scheme with exact findings of spatial derivatives is better than considering linearized equations. The error obtained by the proposed scheme with the determination of exact spatial derivatives is better than that obtained by two explicit existing schemes. The main advantage of the proposed scheme is that it is capable of providing the solution for convection-diffusion equations with compact fourth-order accuracy. Still, the corresponding implicit compact scheme is unable to find the solution to convection-diffusion problems.