Second-order Accuracy In Time Research Articles

Analysis of a Crank–Nicolson fast element-free Galerkin method for the nonlinear complex Ginzburg–Landau equation

A fast element-free Galerkin (EFG) method is proposed in this paper for solving the nonlinear complex Ginzburg–Landau equation. A second-order accurate time semi-discrete system is presented by using the Crank–Nicolson scheme for the temporal discretization, and then a meshless fully discrete system is established by using the EFG method for the spatial discretization. In the proposed EFG method, Nitsche’s technique is used to impose the essential boundary conditions in a weak sense, and the reproducing kernel gradient smoothing integration is used to accelerate the calculation. Theoretical errors for the time semi-discrete system and the fully discrete EFG system are analyzed in detail. Optimal error estimates of the fully discrete Crank–Nicolson EFG method are obtained in both L2 and H1 norms. Numerical results validate the theoretical results and the effectiveness of the method.

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.

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.

On the phase-field algorithm for distinguishing connected regions in digital model

In this paper, we propose a novel model for the discrimination of complex three-dimensional connected regions. The modified model is grounded on the Allen–Cahn equation. The modified equation not only maintains the original interface dynamics, but also avoids the unbounded diffusion behavior of the original Allen–Cahn equation. This advantage enables us to accurately populate and extract the complex connectivity region of the target part. The model is discretized employing a semi-implicit Crank–Nicolson scheme, ensuring second-order accuracy in both time and space. This paper provides a rigorous proof of the unconditional energy stability of our method, thereby affirming the numerical stability and the physical rationality of the solution. We validate the discriminative ability of the proposed model for 3D complex connected regions.

Phase-field based modeling and simulation for selective laser melting techniques in additive manufacturing

In this study, we develop a phase-field model to describe the solid–liquid phase changes, heat conduction phenomena, during the selective laser melting process. This model is based on the variational principle of minimizing the free energy functional. The proposed model integrates the phase-field equation and the energy equation, which are used to capture the dynamical behavior of the interfacial evolution. We use the semi-implicit Crank–Nicolson scheme and central difference to ensure second-order accuracy in time and space. The numerical scheme is unconditional energy stable. This paper rigorously proves the energy stability of the phase-field model of the Selective Laser Melting process, which confirms the numerical stability and the physical rationality of the solution. Various numerical experiments are performed to verify the robustness of our proposal model. This model can effectively simulate the energy transfer and shape structure changes of the products during the selective laser melting manufacturing process, which provides a reliable guarantee for predicting and optimizing the quality and performance of the selective laser melting process additive manufacturing process.

A machine-learning augmented mixed method for simulating α particle transport in inertial confinement fusion

The plasma heating by the α particle transport is a main self-heating source in inertial confinement fusion (ICF) that determines the capsule implosion performance. Due to the high energy of α particle and the high temperature in the ICF capsule hot spot, significant non-equilibrium effect exists and the continuum mechanics breaks down. For the numerical simulation of implosion and charged particle transport, the Boltzmann equation needs to be solved to capture the kinetic effects. However, the 7-dimensional Boltzmann equation, the highly frequent Coulomb collision, and the multi-folded integral stopping power formulation greatly limits the computational efficiency and challenges the computational power. To overcome the high computational cost of high-frequent Coulomb collisions, we propose a mixed collision model according to which the collisions are categorized into low-frequent large-angle collisions and high frequent small-angle grazing collisions. The large-angle collision process is precisely solved based on the Coulomb cross-section. For the highly frequent small-angle grazing, a statistic model is constructed with second-order accuracy in time. The mixed collision model reduces the computational cost of scattering calculation by two orders of magnitude. For the multi-folded integral stopping power formulation, a neural network is used to improve computational efficiency. Based on the proposed algorithm, we develop one-dimensional to three-dimensional module code to directly solve the α particle transport Boltzmann equation. The α transport module code is integrated into the multi-physics LARED-S program. The MC version ICF software is verified by a simulation study of the N210207 and N191110 experiment.

A Structure-Preserving Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at all Mach and Alfvén Numbers

We present a divergence-free semi-implicit finite volume scheme for the simulation of the ideal magnetohydrodynamics (MHD) equations which is stable for large time steps controlled by the local transport speed at all Mach and Alfvén numbers. An operator splitting technique allows to treat the convective terms explicitly while the hydrodynamic pressure and the magnetic field contributions are integrated implicitly, yielding two decoupled linear implicit systems. The linearity of the implicit part is achieved by means of a semi-implicit time linearization. This structure is favorable as second-order accuracy in time can be achieved relying on the class of semi-implicit IMplicit–EXplicit Runge–Kutta (IMEX-RK) methods. In space, implicit cell-centered finite difference operators are designed to discretely preserve the divergence-free property of the magnetic field on three-dimensional Cartesian meshes. The new scheme is also particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, since no explicit numerical dissipation is added to the implicit contribution and the time step is scale independent. Likewise, highly magnetized flows can benefit from the implicit treatment of the magnetic fluxes, hence improving the computational efficiency of the novel method. The convective terms undergo a shock-capturing second order finite volume discretization to guarantee the effectiveness of the proposed method even for high Mach number flows. The new scheme is benchmarked against a series of test cases for the ideal MHD equations addressing different acoustic and Alfvén Mach number regimes where the performance and the stability of the new scheme is assessed.

Accelerated numerical modeling of shallow water flows with MPI, OpenACC, and GPUs

In this paper, a time-explicit Finite-Volume method is adopted to solve the 2-D shallow water equations on an unstructured triangular mesh, using a two-stage Runge-Kutta integrator and a monotone MUSCL model to achieve second-order accuracy in time and space, respectively. A multi-GPU model is presented that uses the Message Passing Interface (MPI) with OpenACC and uses the METIS library to produce the domain decomposition. A CUDA-aware MPI library (GPUDirect) and overlapped MPI communication with computation are used to improve parallel performance. Two benchmark tests with wet and dry downstream beds are used to test the code's accuracy. Good results were achieved compared to the numerical simulations of published studies. Compared with the multi-CPU version of a 6-core CPU, maximum speedups of 56.18 and 331.51 were obtained using a single GPU and 8 GPUs, respectively. Higher mesh resolution enhances acceleration performance, and the model is applicable to other environmental modeling activities.

A uniformly convergent method for two-parameter singularly perturbed parabolic partial differential equations with a large shift

A class of singularly perturbed two-parameter parabolic partial differential equations with a large shift is studied. These equations include time-dependent variables and exhibit sensitivity to small perturbations in the system parameters. Moreover, the shift captures the influence of neighbouring states or events on the current evolution of the system. It adds memory-like behaviour to the problem, making their analysis and numerical solution more intricate. The solution of these equations exhibits a multiscale character. The interior and boundary layers exist across which the solution has a steep gradient. A higher-order accurate numerical method is proposed on a non-uniform mesh to solve the problem. The method combines an upwind difference scheme in space and a Crank-Nicolson scheme in time. The method is unconditionally stable, and parameters uniformly convergent with second-order accuracy in time and first-order accuracy in the space variable. Besides, the paper presents numerical results and illustrations to support theoretical estimates.

A second-order-in-time, explicit approach addressing the redundancy in the low-Mach, variable-density Navier-Stokes equations

A novel algorithm for explicit temporal discretization of the variable-density, low-Mach Navier-Stokes equations is presented here. Recognizing there is a redundancy between the mass conservation equation, the equation of state, and the transport equation(s) for the scalar(s) which characterize the thermochemical state, and that it destabilizes explicit methods, we demonstrate how to analytically eliminate the redundancy and propose an iterative scheme to solve the resulting transformed scalar equations. The method obtains second-order accuracy in time regardless of the number of iterations, so one can terminate this subproblem once stability is achieved. Hence, flows with larger density ratios can be simulated while still retaining the efficiency, low cost, and parallelizability of an explicit scheme. The temporal discretization algorithm is used within a pseudospectral direct numerical simulation which extends the method of Kim, Moin, and Moser for incompressible flow [17] to the variable-density, low-Mach setting, where we demonstrate stability for density ratios up to ∼25.7.

The application of a non-hydrostatic RANS model for simulating irregular wave breaking on a barred and sloping beach

In the current study, a non-hydrostatic two-dimensional vertical (2DV) numerical model with a shock-capturing technique is developed to simulate irregular wave breaking on a barred beach. The complete form of the 2DV Reynolds-averaged Navier-Stokes (RANS) equations is discretized using the finite volume approach. In the spatial discretization phase, a flux limiter function is employed for the velocity advection terms to prevent non-physical oscillations in regions with steep gradients. A novel method has been introduced during the discretization stage, leading to a significant reduction in computational expenses. For temporal discretization, the Leapfrog method, known for its second-order accuracy in time, is employed to address wave-damping issues. Model validation involves a comparison between simulation results and experimental data pertaining to irregular wave breaking, affirming the satisfactory performance of the model. To evaluate the accuracy of the model, the Root Mean Square Error (RMSE) was calculated. The assessment showed that the model is capable of estimating the significant wave height reported in the experiments used with an RMSE between 0.002 and 0.089, and the mean wave periods with an RMSE between 0.061 and 0.252.

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

Herein, we mainly develop a new reduced-dimension method for the unknown finite element (FE) solution coefficient vectors to the elastic wave equation containing the known singular source term, the unknown displacement vector, and the unknown stress tensor matrix. For this end, we need first to develop the Crank-Nicolson (CN) FE (CNFE) method with unconditional stability, unconditional convergence, and second-order time accuracy for the elastic wave equation and analyze the existence, unconditional stability, and errors of the CNFE solutions. After that we employ a proper orthogonal decomposition to lower the dimensionality of the unknown FE solution coefficient vectors in the CNFE method for the elastic wave equation and to design a new reduced-dimension extrapolated CNFE (RDECNFE) method. Subsequently, we employ matrix analysis to discuss the existence, unconditional stability, unconditional convergence, and errors of the RDECNFE solutions. Finally, we use two numerical examples to show the superiority of the RDECNFE method and our theoretical results. It is also shown that the RDECNFE method is feasible and effective for solving the elastic wave equation.

Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources

Abstract Scientific inquiry into effective numerical methods for modelling complex physical processes has led to the investigation of fluid dynamics, mainly when non-Newtonian properties and complex heat sources are involved. This paper presents an enhanced exponential time integrator approach to dynamically simulate non-Newtonian boundary layer flow with spatially and temporally varying heat sources. We propose an explicit scheme with second-order accuracy in time, demonstrated to be stable through Fourier series analysis, for solving time-dependent partial differential equations (PDEs). Utilizing this scheme, we construct and solve dimensionless PDEs representing the flow of Williamson fluid under the influence of space- and temperature-dependent heat sources. The scheme discretizes the continuity equation of incompressible fluid and Navier–Stokes, energy, and concentration equations using the central difference in space. Our analysis illuminates how factors affect velocity, temperature, and concentration profiles. Specifically, we observe a rise in temperature profile with enhanced coefficients of space and temperature terms in the heat source. Non-Newtonian behaviours and geographical/temporal variations in heat sources are critical factors influencing overall dynamics. The novelty of our work lies in developing an explicit exponential integrator approach, offering stability and second-order accuracy, for solving time-dependent PDEs in non-Newtonian boundary layer flow with variable heat sources. Our results provide valuable quantitative insights for understanding and controlling complex fluid dynamics phenomena. By addressing these challenges, our study advances numerical techniques for modelling real-world systems with implications for various engineering and scientific applications.

Convergence analysis of the maximum principle preserving BDF2 scheme with variable time-steps for the space fractional Allen–Cahn equation

In this work, a fully implicit discrete scheme preserving the maximum principle is analyzed for solving the space fractional Allen–Cahn equation with variable step BDF2 in time. The convergence analysis in L∞ norm is rigorously proved under the same requirement of time-step ratios for the maximum principle by means of the kernels recombination technique. Furthermore, the unique solvability of the proposed scheme is theoretically guaranteed depending on the property of convex functions. The energy decay law with the consideration of Riesz fractional derivative is also proved under sufficient restrictions on time-steps and time-step ratios. Numerical experiments are given to validate the second order accuracy in time, the maximum principle as well as the energy decay law of the proposed numerical scheme. In particular, comparisons of errors and time levels for the adaptive time-stepping strategy and uniform time steps are conducted to demonstrate the efficiency of adaptive time-stepping strategy.

Coupled two-dimensional model for heavily sediment-laden floods and channel deformation in a braided reach of the Lower Yellow River

It is difficult to predict the detailed morphodynamic processes induced by heavily sediment-laden floods in alluvial rives, due to the complexity in topography and the significant channel evolution rates, especially in a braided reach of the Lower Yellow River (LYR). A two-dimensional morphodynamic model using a coupled solution approach was firstly developed, with the effects of sediment concentration and bed deformation being directly considered in the hydrodynamic governing equations. The finite volume method with the HLL-MUSCL scheme was adopted in the numerical solution, which can obtain second-order accuracy in space and time. In addition, unstructured meshes were used to accommodate the irregular boundaries. The proposed model was validated using measurements in the braided reach of the LYR during the flood events of 2020 and 2004, with general agreement obtained between the calculated and measured hydrographs of flow and sediment concentration. The highest accuracy was achieved in the simulation of discharge, with a Nash-Sutcliffe efficiency coefficient (NSE) of 0.92. The simulation accuracy in sediment concentration was improved by the coupled approach, with the relative error of the calculated peak value decreasing from 17 % to 9 %. In addition, transport characteristics of graded suspended sediments were well simulated, with the fine fraction being transported downstream and the medium fraction depositing in the study reach. Influences of channel adjustments on the sediment transport processes were investigated during a flood event. The results demonstrated that the incised channel in 2020 was characterized by a larger water depth and smaller velocity, as compared with the shallow channel in 2004. Consequently, the peak discharge decreased and the flood propagation time was extended under the 2020 topography, associated with the lower sediment discharge at the outlet due to the reduced sediment transport capacity.

The Second-Order Numerical Approximation for a Modified Ericksen–Leslie Model

In this study, two numerical schemes with second-order accuracy in time for a modified Ericksen–Leslie model are constructed. The highlight is based on a novel convex splitting method for dealing with the nonlinear potentials, which is integrated with the second-order backward differentiation formula (BDF2) and leap frog method for temporal discretization and the finite element method for spatial discretization. The unconditional energy stability of both schemes is further demonstrated. Finally, several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed schemes.

A Numerical Method Based on Operator Splitting Collocation Scheme for Nonlinear Schrödinger Equation

In this paper, a second-order operator splitting method combined with the barycentric Lagrange interpolation collocation method is proposed for the nonlinear Schrödinger equation. The equation is split into linear and nonlinear parts: the linear part is solved by the barycentric Lagrange interpolation collocation method in space combined with the Crank–Nicolson scheme in time; the nonlinear part is solved analytically due to the availability of a closed-form solution, which avoids solving the nonlinear algebraic equation. Moreover, the consistency of the fully discretized scheme for the linear subproblem and error estimates of the operator splitting scheme are provided. The proposed numerical scheme is of spectral accuracy in space and of second-order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, mass and energy conservation of the proposed method.

Arbitrary-Lagrangian-Eulerian finite volume IMEX schemes for the incompressible Navier-Stokes equations on evolving Chimera meshes

In this article we design a finite volume semi-implicit IMEX scheme for the incompressible Navier-Stokes equations on evolving Chimera meshes. We employ a time discretization technique that separates explicit and implicit terms, accommodating the multi-scale nature of the governing equations, which involve both time scales of diffusion and advection operators. The finite volume approach for both explicit and implicit terms allows to encode into the nonlinear flux the velocity of displacement of the Chimera mesh via integration on moving cells. The numerical solution is then projected onto the physically meaningful solution manifold of non-solenoidal fields that stems from the energy equation. To attain second-order time accuracy, we employ semi-implicit IMEX Runge-Kutta schemes. These novel schemes are combined with a fractional-step method, thus the governing equations are eventually solved using a projection method to satisfy the divergence-free constraint of the velocity field. The implicit discretization of the viscous terms allows the CFL-type stability condition for the maximum admissible time step to be only defined by the relative fluid velocity referred to the movement of the frame and not depending on the viscous terms. Communication between different grid blocks is enabled through compact exchange of information from the fringe cells of one mesh block to the field cells of the other block. The continuity of the solution is recovered in one-shot during the solution of the arising algebraic systems by not involving neither direct discretization of the differential operators on fringe cells nor an iterative Schwartz-type method. Free-stream preservation property, i.e. compliance with the Geometric Conservation Law (GCL), is respected at the order of the scheme. The accuracy and capabilities of the new numerical schemes are proved through an extensive range of test cases, demonstrating ability to solve relevant benchmarks in the field of incompressible fluids.

A space-time second-order method based on modified two-grid algorithm with second-order backward difference formula for the extended Fisher–Kolmogorov equation

In this paper, a modified two-grid algorithm based on block-centred finite difference method is developed for the fourth-order nonlinear extended Fisher–Kolmogorov equation. To further improve the computational efficiency, an effective second-order accurate backward difference formula is considered. The modified two-grid method based on Newton iteration is constructed to linearize the nonlinear system. The method solves a miniature nonlinear system on a coarse grid accompanying a larger time step to get the numerical solution, then computes a linear system constructed by the previous result with the Taylor expansion on a fine grid accompanying a smaller time step to get the correct numerical solution. Theoretical analysis shows that the modified two-grid algorithm can achieve second-order convergence accuracy both in time and space domain. Several numerical experiments are provided to verify the theoretical result and the high efficiency of this approach. The practical problem illustrates the actual applicable value of the algorithm.

A high accuracy compact difference scheme and numerical simulation for a type of diffusive plant-water model in an arid flat environment

<abstract><p>In this paper, we investigate the numerical computation method for a one-dimensional self diffusion plant water model with homogeneous Neumann boundary conditions. First, a high accuracy compact difference scheme for the diffusive plant water model in an arid flat environment is constructed using the finite difference method. The fourth order compact difference scheme is used for the spatial derivative term, and the Taylor series expansion and residual correction function are used to discretize the time term. We obtain a difference scheme with second-order accuracy in time and fourth-order accuracy in space. Second, the Fourier analysis method is used to prove that the above format is unconditionally stable. Then, the numerical examples provided the convergence and accuracy of the difference scheme. Finally, numerical simulations are conducted near the Turing Hopf bifurcation point of the model to obtain the spatial distribution maps of vegetation and water under small disturbances of different parameters. In this paper, the evolution law of vegetation quantity and water density at any time is observed.Revealing the impact of small changes in parameters on the spatiotemporal dynamics of plant water models will provide a basis for understanding whether ecosystems are fragile.</p></abstract>