Time Approximation Research Articles

Semi-implicit schemes for modeling water flow and solute transport in unsaturated soils

The coupled model of water flow and solute transport in unsaturated soils is addressed in this study. Building upon previous research findings by Keita, Beljadid, and Bourgault, we investigate a class of second-order time-stepping techniques where two free parameters are introduced, to identify the most stable and accurate scheme. The spatial discretization of the Richards equation is accomplished using the mixed finite element method. The proposed approach involves formulating noniterative schemes using an extrapolation formula and Taylor approximation in time to linearize nonlinear terms. Additionally, a specialized regularization technique is applied to ensure the convergence of the proposed numerical methods. Numerical simulations are conducted to determine the optimal scheme for solving the Richards equation, which is subsequently extended to the transport equation.Numerical simulations of water flow reveal the good accuracy of three schemes—SBDF, MSBDF, and Richards-M2 for homogeneous and heterogeneous soils. Notably, the SBDF scheme stands out for its computational efficiency and stability, especially when gravity forces dominate over capillary forces. Through numerical analysis of the coupled semi-implicit schemes, our results affirm the SBDF scheme’s superior robustness, establishing it as the optimal choice among the proposed numerical methods. Therefore, the SBDF scheme is employed to solve the coupled model of water flow and solute transport. We conducted various numerical experiments to solve the coupled model, addressing scenarios including single and multispecies nitrogen transport, pore water electrical conductivity, and nitrate transport. The SBDF scheme’s accuracy was rigorously verified through comparisons with reference solutions and experimental data. This establishes the SBDF scheme as an efficient alternative to traditional implicit methods for modeling water flow and solute transport in unsaturated soils.

Numerical solutions of Schrödinger–Boussinesq system by orthogonal spline collocation method

This paper is concerned with the numerical solutions of Schrödinger–Boussinesq (SBq) system by an orthogonal spline collocation (OSC) discretization in space and Crank–Nicolson (CN) type approximation in time. By using the mathematical induction argument and standard energy method, the proposed CN+OSC scheme is proved to be unconditionally convergent at the order O(τ2+h4) with mesh-size h and time step τ in the discrete L2-norm. We devise a new computation method based on the orthogonal diagonalization techniques (ODT) to realize the proposed CN+OSC scheme. In order to compare the performance of ODT, we devise an alternating direction implicit (ADI) method to compute the CN+OSC scheme for high spatial dimension SBq system. As an alternative implementation, the new method ODT not only exhibits more accurate numerical results, but also demonstrates stronger invariance preserving ability. Numerical results are reported to verify the error estimates and the discrete conservation laws.

Renewal of the regression model for normalization of specific energy consumption

Purpose. To develop a method of updating the regression model for the normalization of specific energy consumption in the presence of frequent and significant changes in the energy efficiency of the production process. Methodology. Analysis of existing methods of updating regression models, comparison of their possibilities, and synthesis of the method of updating the model in conditions of frequent and significant changes in the energy efficiency of the production process. Findings. It was established that in the presence of a significant number of possible variants of structural and mode changes in the energy consumption of the control object, the introduction of associated variables into the regression model is problematic, as it requires an increase in the number of experimental data in conditions of their expected heterogeneity. The flaw of the well-known regression model for normalizing the power consumption of the object of control is revealed, which consists of the fact that the model does not take into account the last values in the sequence of their appearance of experimental data obtained in the process of energy efficiency control. This reduces the accuracy of predicted energy consumption values. It is proposed to update the regression model every time after performing the energy efficiency control and sample adjustment. Adjustments are implemented by checking the homogeneity of the obtained experimental data, followed by their addition to the elements of the existing sample and removal (if necessary) from the sample of outdated data. The defined sequence of adjustment of the initial data allows timely updating of the model and implementation of the forecast of specific energy consumption, entering data reflecting the latest changes that occurred in the facility's energy supply. The proposed method of updating the model implements the approximation in time of the moment of energy efficiency control to the moment of obtaining experimental data for building a regression dependence for normalizing energy consumption values. This helps to increase the accuracy of the forecast of normalized values. A significant change in the conditions of production of products with a violation of the homogeneity of data is accompanied by a transition to the transitional mode of adjustment, where it is proposed to reduce the number of elements of the existing sample, ensuring the sequential removal of the elements furthest from the next moment of control. Extraction continues until data homogeneity is achieved. During the daily control of the efficiency of electricity consumption, the change in the values of the regression model coefficients in the process of its renewal reflects the changes in the object's electricity consumption that occurred over the last day. This allows you to separate their impact from the impact of changes that occurred earlier and to assess the level of this impact. Originality. For the first time, the shortcomings of the existing methods of updating regression models in the conditions of frequent and significant changes in the energy efficiency of the production process were identified. A method of updating the model under these conditions has been developed, which involves adjusting the sample of experimental data by changing the number of its elements and checking the homogeneity of the data. Practical value is that the sequence of actions during the implementation of the developed method of updating the regression model is defined, which allows for an increase in the accuracy of calculating the normalized values of specific energy consumption.

Efficient spectral and spectral element methods for Sobolev equation with diagonalization technique

In this paper, we first introduce a new series of Legendre basis functions by using the matrix decomposition technique, which are simultaneously orthogonal in both L2- and H1-inner products. Then, we construct efficient space-time spectral method for linear Sobolev equation using spectral approximation in space and multi-domain collocation approximation in time, which can be implemented in a synchronous parallel fashion. Next, we propose a novel Legendre spectral element method for solving nonlinear Sobolev equation with Burgers' type term, which reduce the non-zero entries of linear systems and computational cost. Some rigorous error estimates are carried out for one-dimensional Sobolev equation. Numerical experiments illustrate the effectiveness and accuracy of the suggested approaches.

Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial Diameter

The fundamental Sparsest Cut problem takes as input a graph G together with edge capacities and demands and seeks a cut that minimizes the ratio between the capacities and demands across the cuts. For n -vertex graphs G of treewidth k , Chlamtáč, Krauthgamer, and Raghavendra (APPROX’10) presented an algorithm that yields a factor- \(2^{2^k}\) approximation in time \(2^{O(k)} \cdot n^{O(1)}\) . Later, Gupta, Talwar, and Witmer (STOC’13) showed how to obtain a 2-approximation algorithm with a blown-up runtime of \(n^{O(k)}\) . An intriguing open question is whether one can simultaneously achieve the best out of the aforementioned results, that is, a factor-2 approximation in time \(2^{O(k)} \cdot n^{O(1)}\) . In this article, we make significant progress towards this goal via the following results: (i) A factor- \(O(k^2)\) approximation that runs in time \(2^{O(k)} \cdot n^{O(1)}\) , directly improving the work of Chlamtáč et al. while keeping the runtime single-exponential in k . (ii) For any \(\varepsilon \in (0,1]\) , a factor- \(O(1/\varepsilon ^2)\) approximation whose runtime is \(2^{O(k^{1+\varepsilon }/\varepsilon)} \cdot n^{O(1)}\) , implying a constant-factor approximation whose runtime is nearly single-exponential in k and a factor- \(O(\log ^2 k)\) approximation in time \(k^{O(k)} \cdot n^{O(1)}\) . Key to these results is a new measure of a tree decomposition that we call combinatorial diameter , which may be of independent interest.

A structure-preserving upwind DG scheme for a degenerate phase-field tumor model

In this work, we present a modification of the phase-field tumor growth model given in [1] that leads to bounded, more physically meaningful, volume fraction variables. In addition, we develop an upwind discontinuous Galerkin (DG) scheme preserving the mass conservation, pointwise bounds and energy stability of the continuous model. Finally, some computational tests in accordance with the theoretical results are introduced. In the first test, we compare our DG scheme with the finite element (FE) scheme related to the same time approximation. The DG scheme shows a well-behavior even for strong cross-diffusion effects in contrast with FE where numerical spurious oscillations appear. Moreover, the second test exhibits the behavior of the tumor-growth model under different choices of parameters and also of mobility and proliferation functions.

Three-Level Schemes with Double Change in the Time Step

Nonstationary problems are solved numerically by applying multilevel (with more than two levels) time approximations. They are easy to construct and relatively easy to study in the case of uniform grids. However, the numerical study of application-oriented problems often involves approximations with a variable time step. The construction of multilevel schemes on nonuniform grids is associated with maintaining the prescribed accuracy and ensuring the stability of the approximate solution. In this paper, three-level schemes for the approximate solution of the Cauchy problem for a second-order evolution equation are constructed in the special case of a doubled or halved step size. The focus is on the approximation features in the transition between different step sizes. The study is based on general results of the stability (well-posedness) theory of operator-difference schemes in a finite-dimensional Hilbert space. Estimates for stability with respect to initial data and the right-hand side are obtained in the case of a doubled or halved time step.

A deep learning method for the dynamics of classic and conservative Allen-Cahn equations based on fully-discrete operators

The Allen-Cahn equation is a well-known stiff semilinear parabolic equation used to describe the process of phase separation and transition in phase field modeling of multi-component physical systems, while the conservative Allen-Cahn equation is a modified version of the classic Allen-Cahn equation that can additionally conserve the mass. As neural networks and deep learning techniques have achieved significant successes in recent years in scientific and engineering applications, there has been growing interest in developing deep learning algorithms for numerical solutions of partial differential equations. In this paper, we propose a novel deep learning method for predicting the dynamics of the classic and conservative Allen-Cahn equations. Specifically, we design two special convolutional neural network models, one for each of the two equations, to learn the fully-discrete operators between two adjacent time steps. The loss functions of the two models are defined using the residual of the fully-discrete systems, which result from applying the central finite difference discretization in space and the Crank–Nicolson approximation in time. This approach enables us to train the models without requiring any ground-truth data. Moreover, we introduce an effective training strategy that automatically generates useful samples along the time evolution to facilitate training of the models. Finally, we conduct extensive experiments in two and three dimensions to demonstrate outstanding performance of our proposed method, including its dynamics prediction and generalization ability under different scenarios.

Analysis of two spectral Galerkin approximation schemes for solving the perturbed FitzHugh-Nagumo neuron model

In this paper, we mainly study two spectral Galerkin approximation schemes of the FitzHugh-Nagumo (FHN) neuron model for the nerve impulse propagation, which contains a small parameter perturbation and strong nonlinearity. To this end, we consider two kinds of temporal discretization by a semi-implicit Euler method and a Crank-Nicolson method, and the spatial discretization by the spectral Galerkin method for the perturbed FHN neuron model. In particular, in order to obtain the second-order approximation in time for the numerical scheme, we modify the nonlinear term f(un) in the Crank-Nicolson spectral Galerkin (CNSG) scheme. We derive the error estimations with optimal convergence rates of the numerical solutions by the first-order semi-implicit spectral Galerkin (SISG) approximation scheme and the second-order modified CNSG (MCNSG) approximation scheme for the perturbed FHN neuron model. Finally, some numerical examples of one-dimensional and two-dimensional nonlinear FHN models with periodic boundary conditions are carried out to verify that the approximate solutions of the proposed SISG and MCNSG schemes satisfy the proved theoretical results.

Linearized fast time-stepping schemes for time–space fractional Schrödinger equations

This paper proposes a linearized fast high-order time-stepping scheme to solve the time–space fractional Schrödinger equations. The time approximation is done by using the fast L2-1σ formula on nonuniform meshes. The spatial discretization is done by using Fourier spectral methods. Optimal error estimates of the numerical method are obtained unconditionally. Such unconditional convergence results are proved directly under the assumption f∈C1(R) by the fractional Sobolev inequalities and boundedness of the numerical solutions. In contrast, the previous unconditionally convergence results are usually proved by using the error splitting argument under the assumption f∈C3(R) and the help of some additional systems. Numerical experiments are given to confirm the theoretical results.

Building of a Mathematical Model for Solving the Elastic–Gas-Dynamic Task of the Gas Lubrication Theory for Petal Bearings

Petal gas bearings are widely used in industrial turbine installations. However, there are practically no mathematical models that take into account the nonlinear dependence of the elastic element deformations of the gas-dynamic bearing on the pressure. It is very difficult to obtain a converged solution with a given accuracy in an acceptable time. On this basis, a mathematical model of the motion of a gas-dynamic bearing with a petals package was proposed. The model takes into account the pressure dependence of the elastic element deflections of the bearing, as well as the bearing design features and its elastic elements. A feature of solving the presented task was the joint solution of two subtasks: gas lubrication and deformations of the elastic elements of the bearing. The problem was that the contact areas of the elastic elements are not known in advance. The search for deflections or the formulation of the elasticity problem was based on the Lagrange variational principle. Petals are in the shape of thin cylindrical shells. The solution was achieved by minimizing the potential energy of the system of deformed petal shells using the first-order gradient method. The solution to the gas-dynamics problem was achieved by applying an explicit finite-difference approximation in time. To implement the proposed model, a numerical algorithm was developed to solve the related tasks of dynamics and elasticity by combining the methods of variational calculus, optimization, and an explicit finite-difference time scheme. The order of accuracy of convergence of the solution corresponded to 10−5. The study demonstrated that the load-carrying capability of the bearing increased by 2–4 times with an increase in the number of petals. The results of experimental studies allowed us to estimate the interval for the ascent speed of the rotor, which was used as the initial conditions for numerical modeling.

Pressure robust SUPG-stabilized finite elements for the unsteady Navier–Stokes equation

Abstract In the present contribution, we propose a novel conforming finite element scheme for the time-dependent Navier–Stokes equation, which is proven to be both convection quasi-robust and pressure robust. The method is built combining a ‘divergence-free’ velocity/pressure couple (such as the Scott–Vogelius element), a discontinuous Galerkin in time approximation and a suitable streamline upwind Petrov–Galerkin-curl stabilization. A set of numerical tests, in accordance with the theoretical results, is included.

Higher-Order Accurate Explicit Time Schemes with Improved Dissipation Properties

In this paper, a novel family of fourth-order accurate explicit time integration schemes is developed by combining the new time approximations and the explicit fourth-order Runge–Kutta (RK4) method. Inaccurate predictions due to the presence of excessive numerical dissipation are often observed in practical analyses of structural dynamics when the RK4 is employed for both displacement and velocity approximations. To remedy this, novel time approximations with adjustable algorithmic parameters are employed for the displacement vectors while the velocity vectors are approximated by using the RK4. For the complete elimination of numerical dissipation, algorithmic parameters are unconventionally determined by taking the determinant of the amplification matrix as unity. A set of algorithmic parameters obtained from this process makes the new schemes completely non-dissipative while keeping the computational cost the same as the RK4. Due to this improvement, the new schemes have enhanced total energy-conserving capabilities for nonlinear systems and give noticeably more accurate predictions in practical analyses. Until now, controllable numerical dissipation and fourth-order accuracy are not attained simultaneously in a unified set of time schemes. In the new schemes, however, a systematic way to adjust the level of numerical dissipation is also presented while attaining fourth-order accuracy. The numerical results of various test problems show that the new time schemes can provide more accurate predictions for nonlinear conservative dynamic problems than the existing time schemes.

The influence of vacancy generation on the impurity distribution in the target under the conditions of surface treatment by a particle beam

AbstractThe paper presents a nonlinear coupled isothermal model of the process of surface treatment by a particle beam. The model takes into account the interaction between the diffusion of impurity and propagation of mechanical disturbances, as well as the transfer of the introduced impurity with the vacancy mechanism and under the action of stresses. The nonlinear problem is solved numerically using an implicit symmetric difference scheme of the second order approximation in time and spatial coordinates. The finite time of mass flux relaxation and the dependence of diffusion coefficient on the composition and on the concentration of vacancies lead to peculiarities in the propagation of both the concentration wave and the wave of mechanical disturbances. Vacancy generation contributes to the acceleration of impurity propagation and increased strain/stresses.

An Efficient Finite-Difference Stencil with High-Order Temporal Accuracy for Scalar Wave Modeling

Solving a scalar wave equation by the finite-difference (FD) method is a key step for advanced seismic imaging, in which the numerical accuracy is significantly affected by the FD stencil. High-order spatial and temporal approximations of the FD stencil can effectively improve the numerical accuracy and mitigate dispersion error. However, the huge costs of high-order stenciling in computation and storage hinder the application of large-scale modeling. In this paper, we propose a new efficient FD stencil with high-order temporal accuracy for numerical seismic modeling. The new stencil has a radial shape, including a standard cross-stencil and a rotated cross-stencil with a (π/4) degree, and it can reach sixth-order accuracy in the time approximation. Compared with the well-known temporal high-order cross-rhombus stencil, the new stencil involves fewer grid nodes and thus has higher computational efficiency, especially in high-order cases. Dispersion and stability analyses show that the new stencil has great improvements in mitigating the dispersion error and stability problem compared with the conventional methods. Numerical accuracy and execution time analyses show that the new stencil is an economical and feasible method for large-scale modeling.

An unconditionally stable finite element scheme for anisotropic curve shortening flow

Based on a recent novel formulation of parametric anisotropic curve shortening flow, we analyse a fully discrete numerical method of this geometric evolution equation. The method uses piecewise linear finite elements in space and a backward Euler approximation in time. We establish existence and uniqueness of a discrete solution, as well as an unconditional stability property. Some numerical computations confirm the theoretical results and demonstrate the practicality of our method.

A primal-dual approach for solving conservation laws with implicit in time approximations

In this work, we propose a novel framework for the numerical solution of time-dependent conservation laws with implicit schemes via primal-dual hybrid gradient methods. We solve an initial value problem (IVP) for the partial differential equation (PDE) by casting it as a saddle point of a min-max problem and using iterative optimization methods to find the saddle point. Our approach is flexible with the choice of both time and spatial discretization schemes. It benefits from the implicit structure and gains large regions of stability, and overcomes the restriction on the mesh size in time by explicit schemes from Courant–Friedrichs–Lewy (CFL) conditions (really via von Neumann stability analysis). Nevertheless, it is highly parallelizable in principle, and easy-to-implement. In particular, no nonlinear inversions are required! Specifically, we illustrate our approach using the finite difference scheme and discontinuous Galerkin method for the spatial scheme; backward Euler and backward differentiation formulas for implicit discretization in time. Numerical experiments illustrate the effectiveness and robustness of the approach. In future work, we will demonstrate that our idea of replacing an initial-value evolution equation with this primal-dual hybrid gradient approach has great advantages in many other situations.

An over-penalized weak Galerkin method for parabolic interface problems with time-dependent coefficients

Based on the idea of classical discontinuous Galerkin and weak Galerkin finite element methods, we introduce an over-penalized term in the new scheme as a part of stabilization for solving parabolic interface problems. From the double-valued functions defined on interior edges of elements, it is natural to generate jumps of the over-penalized term. An over-penalized weak Galerkin (OPWG) finite element method can be applied very well to interface problems with general imperfect interface. Importantly, the diffusion coefficients of the interface problems depend on both temporal and spacial variables, not only limited in space as usual. With the use of (Pk(K),Pk−1(e),[Pk−1(K)]d) elements, semi-discrete and fully discrete schemes with backward Euler approximation in time are presented, and then the semi-discrete one is analyzed to be unconditionally stable. To analyze error estimates of semi-discrete and fully discrete schemes directly by error equation, we can just have optimal convergence order in energy norm. By virtue of the introduction of an elliptic projection operator, optimal error estimates of those schemes in L2 norm can be proved. Numerical examples are given to validate the efficiency and optimal convergence orders of the new schemes.

Numerical dissipation control of a hybrid large-particle method in vortex instability problems

Current trends in the development of numerical schemes are associated with a decrease in dissipative and dispersion errors, as well as an improvement in the grid convergence of the solution. Achieving the computational properties is not an easy problem since a decrease in scheme viscosity is often associated with an increase in the oscillations of gas dynamic parameters. The paper presents a study of the issues of numerical dissipation control in gas dynamics problems in order to increase the resolution in numerical reproduction of vortex instability at contact boundaries. To solve this problem, a hybrid large-particle method of the second order of approximation in space and time on smooth solutions is used. The method is constructed with splitting by physical processes in two stages: gradient acceleration and deformation of the finite volume of the medium; convective transfer of the medium through its facets. An increase in the order of approximation in time is achieved by a time correction step. The regularization of the numerical solution of problems at the first stage of the method consists in the nonlinear correction of artificial viscosity which, regardless of the grid resolution, tends to zero in the areas of smoothness of the solution. At the stage of convective transport, the reconstruction of fluxes was carried out by an additive combination of central and upwind approximations. A mechanism for regulating the numerical dissipation of the method based on a new parametric limiter of artificial viscosity is proposed. The optimal adjustment of the method by the ratio of dissipative and dispersive properties of the numerical solution is achieved by setting the parameter of the limiting function. The efficiency of the method was tested on two-dimensional demonstrative problems. In one of them, the contact surfaces are twisted into a spiral on which the Kelvin-Helmholtz vortex instability develops. Another task is the classic problem with a double Mach reflection of a strong shock wave. Comparison with modern numerical schemes has shown that the proposed variant of the hybrid large-particle method has a high competitiveness. For example, in the problem with double Mach reflection, the considered version of the method surpasses in terms of vortex resolution the popular WENO (Weighted Essentially Non-Oscillatory) scheme of the fifth order and is comparable to the numerical solution of WENO of the ninth order of approximation. The proposed method can be the basis of a convective block of a numerical scheme when constructing a computational technology for modeling turbulence.

Gas kinetic principles in Navier–Stokes finite-volume solvers

We present two simplified gas kinetic models for the computation of the Navier–Stokes equations for compressible flows. Upon starting from the non-linear BGK kinetic model, we derive a simplified procedure for computing the normalized particle velocity distribution function, ψ.From a first-order approximation in time and a directional splitting of the BGK equation, the function ψ is computed over each time-step and at each cell interface of a finite-volume algorithm, as a convex combination between its value at the beginning of each time-step and an equilibrium state of the gas.Hence, depending on the definition of this equilibrium state, we can generate two kinetic theory inspired schemes.The first scheme (SCG scheme), is designed by computing the initial equilibrium state from two Maxwellian distributions of the gas, at each cell interface. The second scheme (ASCGE scheme) is designed by introducing acoustic principles into the definition of this initial equilibrium state. Then, the resulting flux function for convective terms is computed from the gas distribution function, ψ, at each time-step.Viscous fluxes are weakly coupled with convective fluxes by using kinematic features coming from both sides of the generic cell face.Finally, these convective and diffusive fluxes are introduced into a fourth-order MUSCL-like finite-volume procedure for modeling compressible flows.These two resulting simplified gas kinetic models are tested and compared with an HLLC Riemann solver scheme and recent simplified gas kinetic schemes over a variety of physical problems including inviscid/viscous and steady/unsteady problems for a compressible flow.