Temporal Discretization Schemes Research Articles

The energy-diminishing weak Galerkin finite element method for the computation of ground state and excited states in Bose-Einstein condensates

In this paper, we employ the weak Galerkin (WG) finite element method and the imaginary time method to compute both the ground state and the excited states in Bose-Einstein condensate (BEC) which is governed by the Gross-Pitaevskii equation (GPE). First, we use the imaginary time method for GPE to get the nonlinear parabolic partial differential equation. Subsequently, we apply the WG method to spatially discretize the parabolic equation. This yields a semi-discrete scheme, in which an energy function is explicitly defined. For the case β⩾0, we demonstrate that the energy is diminishing with respect to time t at each time step. Applying the backward Euler scheme for temporal discretization yields a fully discrete scheme. For the case β=0, we provide a mathematical justification, establishing the convergence analysis for the numerical solution of the ground state. Moreover, based on the theory of solving eigenvalue problems using the WG method, we present the error estimates between the ground state and its numerical solution under the H1 and L2 norms. Numerical experiments are provided to illustrate the effectiveness of the proposed schemes. Moreover, the results indicate that our method also can compute the first excited state, achieving optimal convergence orders.

A radial basis function-finite difference method for solving Landau–Lifshitz–Gilbert equation including Dzyaloshinskii-Moriya interaction

This paper investigates a numerical method for solving the two-dimensional Landau–Lifshitz–Gilbert (LLG) equation, governing the dynamics of the magnetization in ferromagnetic materials. Specifically, we incorporate the Dzyaloshinskii–Moriya interaction into the LLG equation—a crucial factor for the creation and stabilization of magnetic skyrmions. We propose a local meshless method that utilizes radial basis function-finite difference (RBF-FD) for spatial discretization and the Crank–Nicolson scheme for temporal discretization, along with an extrapolation technique to handle the nonlinear terms. We demonstrate the method’s accuracy, efficiency, and adaptability through numerical tests on domains of various shapes, showcasing its practical utility in simulating real-world magnetic phenomena and advanced materials.

Numerical analysis of nonlinear Klein–Gordon equations by a meshless superconvergent finite point method

In this paper, we introduce a meshless method for numerical simulation of nonlinear Klein–Gordon equations. The method begins with a temporal discretization to address time derivatives. The stability and error of the temporal discretization scheme are theoretically analyzed. Subsequently, meshless algebraic systems of Klein–Gordon solitons are established by using the superconvergent finite point method (SFPM) for spatial discretization. The moving least squares approximation and its smoothed derivatives are adopted in the SFPM to ensure the high accuracy and remarkable superconvergence. Accuracy and convergence of the meshless numerical simulation for nonlinear Klein–Gordon equations are analyzed in theory. Numerical results validate the superconvergence and effectiveness of the method and confirm the theoretical analysis.

An artificial compressibility SAV finite element method for the time-dependent natural convection problem

In this article, we present, analyze, and test a first-order implicit-explicit type scheme based on the artificial compressibility (AC) method and the scalar auxiliary variable (SAV) approach for solving the time-dependent natural convection problem. The proposed method is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization, and explicit treatment for nonlinear terms. It effectively decouples the velocity and pressure via artificial compressibility, thereby reducing computational complexity and execution time. Moreover, we use the SAV approach for the convective terms, it leads to the scheme is linear, and only require solving a sequence of linear differential equation with constant coefficients at each time step. It is proved that the solution of this method is bounded and satisfying the law of energy dissipation. The rigorous error estimate of velocity, pressure, and temperature are derived. Finally, some numerical tests are implemented to verify the theoretical analysis and illustrate the efficiency of the presented method.

Analysis of a meshless generalized finite difference method for the time-fractional diffusion-wave equation

In this paper, a generalized finite difference method (GFDM) is proposed and analyzed for meshless numerical solution of the time-fractional diffusion-wave equation. Two (3−α)-order accurate temporal discretization schemes are presented by using the L1 formula and the original H2N2 or fast H2N2 formulas to discretize the time-fractional derivative of order α∈(1,2). The stability of the temporal discretization schemes is analyzed. Then, the time-fractional diffusion-wave initial-boundary value problem is transformed into a series of time-independent integer-order boundary value problems, and discrete linear algebraic systems are built by the application of the GFDM. Accuracy analysis of the GFDM with both original H2N2 and fast H2N2 formulas is presented in theory, and numerical experimental results are provided to verify the theoretical results and the effectiveness of the proposed meshless method.

Mathematical study of a new coupled electro-thermo radiofrequency model of cardiac tissue

This paper presents a nonlinear reaction–diffusion-fluid system that simulates radiofrequency ablation within cardiac tissue. The model conveys the dynamic evolution of temperature and electric potential in both the fluid and solid regions, along with the evolution of velocity within the solid region. By formulating the system that describes the phenomena across the entire domain, encompassing both solid and fluid phases, we proceed to an analysis of well-posedness, considering a broad class of right-hand side terms. The system involves parameters such as heat conductivity, kinematic viscosity, and electrical conductivity, all of which exhibit nonlinearity contingent upon the temperature variable. The mathematical analysis extends to establishing the existence of a global solution, employing the Faedo–Galerkin method in a three-dimensional space. To enhance the practical applicability of our theoretical results, we complement our study with a series of numerical experiments. We implement the discrete system using the finite element method for spatial discretization and an Euler scheme for temporal discretization. Nonlinear parameters are linearized through decoupling systems, as introduced in our continuous analysis. These experiments are conducted to demonstrate and validate the theoretical findings we have established.

Inconsistencies in unstructured geometric volume-of-fluid methods for two-phase flows with high density ratios

Geometric flux-based Volume-of-Fluid (VOF) methods (Marić et al., 2020) are widely considered consistent in handling two-phase flows with high density ratios. However, although the conservation of mass and momentum is consistent for two-phase incompressible single-field Navier–Stokes equations without phase-change (Liu et al., 2023), discretization may easily introduce inconsistencies that result in very large errors or catastrophic failure. We apply the consistency conditions derived for the ρLENT unstructured Level Set/Front Tracking method (Liu et al., 2023) to flux-based geometric VOF methods (Marić et al., 2020), and implement our discretization into the plicRDF-isoAdvector geometrical VOF method (Roenby et al., 2016). We find that computing the mass flux by scaling the geometrically computed fluxed phase-specific volume can ensure equivalence between the mass conservation equation and the phase indicator (volume conservation) if consistent discretization schemes are chosen for the temporal and convective term. Based on the analysis of discretization errors, we suggest a consistent combination of the temporal discretization scheme and the interpolation scheme for the momentum convection term. We confirm the consistency by solving an auxiliary mass conservation equation with a geometrical calculation of the face-centered density (Liu et al., 2023). We prove the equivalence between these two approaches mathematically and verify and validate their numerical stability for density ratios within [1, 106] and viscosity ratios within [102, 105].

A mixed finite element spatial discretization scheme and a higher‐order accurate temporal discretization scheme for a strongly discontinuous electromagnetic interface condition in ideal sliding electrical contact problems

AbstractSliding electrical contact involves multiple conductors sliding in contact at different speeds, with current flowing through the contact surfaces. The Lagrangian method is commonly used to describe the electromagnetic field in order to overcome the trouble of convective dominance, especially in high‐speed sliding electrical contact problems. However, to maintain correct field continuity, magnetic vector potential and scalar potential taken as variables cannot be continuous simultaneously at the ideal sliding electrical contact interface. This involves a strongly discontinuous condition for variables. Further, commonly used spatial–temporal discretization algorithms are invalid, for example, the classic finite element (CFEM) framework does not allow discontinuous variables, and the backward Euler method in time domain introduces a significant interface error source associated with the velocity of relative motion. To accurately handle strongly discontinuous conditions in numerical calculations, a mixed nodal finite element scheme and a higher‐order accurate temporal discretization scheme are introduced. In this scheme, classical finite element method is performed in each subdomain, and the derivative terms at the boundary are added as new variables. The effectiveness and accuracy of the above methods are verified by comparing them with a standard solution in a two‐dimensional railgun model and analyzing the current density distribution in a three‐dimensional railgun model.

Meshless analysis of fractional diffusion-wave equations by generalized finite difference method

In this paper, a meshless generalized finite difference method (GFDM) is proposed to solve the time fractional diffusion-wave (TFDW) equations. A second-order temporal discretization scheme is developed to tackle the Caputo fractional derivative, and then spatial discretization formulas are derived by the GFDM. Theoretical accuracy and convergence of the GFDM for TFDW equations are analyzed. Numerical results verify the theoretical results and the efficiency of the method.

Numerical investigation of mixing and heat transfer in a 7.9 m JP-8 pool fire

The response of objects engulfed in, and adjacent to, large-scale pool fires is of interest in accident and safety assessments. In this study, Fuego, a low-Mach turbulent reacting flow code was used to study conjugate heat transfer in a 7.9 m diameter JP-8 pool fire. Simulations were designed to replicate past experimental measurements (Blanchat et al., 2006) of incident heat flux to three cylindrical calorimeters in and around the pool fire. Two turbulent combustion models were compared directly - the eddy dissipation concept and a more recently developed unsteady flamelet model. First and second order spatial and temporal discretization schemes were also compared to assess the performance of low-dissipation numerical operators. Heat flux predictions to the transportation size calorimeter outside the fire were within experimental uncertainties. Inside the fire, experimental measurements were higher than predicted values and may have been a consequence of soot deposition and augmented participating media radiation from soot and fuel vapor. Simulation predictions improved in cases where turbulent kinetic energy and mixing were more resolved. This work, and others referenced herein, suggest that spatial resolution on the order of 0.5–1.0 cm may be required to fully resolve fluid instabilities, vortex production between fire plumes and crosswind, soot production, and fuel-air mixing. This presents a substantial computational challenge for safety assessments of engulfed objects in fully turbulent pool fires.

Analysis of a fast element-free Galerkin method for the multi-term time-fractional diffusion equation

In this paper, a fast element-free Galerkin (EFG) method is proposed for solving the multi-term time-fractional diffusion equation (TFDE). Through the use of the multi-term L2−1σ formula to discrete the multi-term Caputo time-fractional derivative, a fast second-order scheme is presented for the temporal discretization, and then the EFG method is used to perform the spatial discretization. The stability of the temporal discretization scheme is discussed. By verifying the fractional Grönwall inequality, the global error of the proposed EFG method for the multi-term TFDE is analyzed in theory. Numerical results validate the theoretical results and the effectiveness of the method.

Numerical Analysis and Computation of a Subgrid-Sparse-Grad-Div Stabilization Method for Incompressible Flow Problems

In this article, a subgrid-sparse-grad-div method for incompressible flow problem was proposed, which is a combination of the subgrid stabilization method and the recently proposed sparse-grad-div method. The method maintains the advantage of both methods: (i) It is robust for solving incompressible flow problem with dominance of the convection, especially when the viscosity is too small. (ii) It can keep mass conservation. Therefore, the method is very efficient for solving incompressible flow. Moreover, based on the Crank–Nicolson extrapolated scheme for temporal discretization, and mixed finite element in spatial discretization, we derive the unconditional stability and optimal convergence of the method. Finally, numerical experiments are proposed to validate the theoretical predictions and demonstrate the efficiency of the method on a test problem for incompressible flow.

Unconditional error analysis of linearized BDF2 mixed virtual element method for semilinear parabolic problems on polygonal meshes

In this paper, we construct, analyze, and numerically validate a class of H(div)-mixed virtual element method for the semilinear parabolic problem in mixed form, in which the parabolic problem is reformulated in terms of the velocity and the pressure of the time-dependent Darcy flow. The Newton linearized method for the nonlinear term is designed to cooperate with the second-order backward differentiation formula of the temporal discretization scheme. This allows each time step to only require the solution of a small and well-structured linear system rather than the solution of a nonlinear system. The linearization improves computational efficiency without decreasing convergence rates. Moreover, the “Fortin” operator and its approximation properties are applied to derive an optimal error estimates O(hk+1+τ2) for the virtual element solution of the velocity and the pressure. Finally, its remarkable performance is illustrated by several numerical examples that also validate the theoretical rates of convergence.

A robust weak Galerkin finite element method for two parameter singularly perturbed parabolic problems on nonuniform meshes

In this study, we proposed a weak Galerkin finite element method (WG-FEM) for solving two-parameter singularly perturbed parabolic problems (TP-SPPPs) of convection–diffusion–reaction type on nonuniform mesh. The WG-FEM approach incorporates an interpolation operator IN and achieves optimal order convergence in an energy-like norm for continuous and fully discrete schemes. In the fully-discrete analysis, we combined WG-FEM for spatial space on a Shishkin-type mesh with the Crank–Nicolson scheme for temporal discretization on an equidistant mesh. Since the energy norm is insufficient to capture the behavior of the boundary layer functions accurately, we have derived the optimal order of convergence in a strong balanced norm. This norm exhibits an order of O(N−1lnN)k in spatial convergence and second-order convergence in time when using an L2-projection QN as an intermediate operator. We conducted numerical examples to validate the proposed method’s theoretical findings. The results of these experiments confirmed the theoretical conclusions and demonstrated the robustness of the proposed method.

Numerical Study of the Effect of Primary Nozzle Geometry on Supersonic Gas-Solid Jet of Bypass Injected Dry Powder Fire Extinguishing Device

A two-way coupled model between polydisperse particle phases with compressible gases and a density-based coupling implicit solution method, combining the third-order MUSCL with QUICK spatial discretization scheme and the second-order temporal discretization scheme, are constructed based on the discrete-phase model (DPM) and the stochastic wander model (DRWM) in the Eulerian–Lagrangian framework in conjunction with a unitary particulate source (PSIC) approach and the SST k-ω turbulence model. The accuracy of the numerical prediction method is verified using previous supersonic nozzle gas-solid two-phase flow experiments. Numerical simulation of a two-phase jet of dry powder extinguishing agent gas with pilot-type supersonic nozzle was performed to analyze the influence of geometrical parameters, such as the length ratio rL and the area ratio rA of the main nozzle on the two-phase flow field, as well as on the jet performance indexes, such as the particle mean velocity vp,a, velocity inhomogeneity Φvp, particle dispersion Ψp, particle mean acceleration ap,a, etc. By analyzing the parameters, we indicate the requirements for the combination of jet performance metrics for different flame types such as penetrating, spreading, and dispersing.

Development of a Fully-Implicit Second-Order system Thermal-Hydraulics analysis code for reactor modeling and simulations

There has been continuous advancement in numerical techniques and software engineering in the field of system thermal-hydraulics over the past decade. The current state-of-the-art involves the use of fully-implicit high-resolution schemes in conjunction with the Jacobian-Free Newton-Krylov (JFNK) method for both single-phase and two-phase flows. However, the application of these advanced schemes in system thermal-hydraulics codes is not a common practice. Given the opportunity and the need to develop a new modern system thermal–hydraulic code from scratch, there is no doubt that these recent advancements should be leveraged. In this work, we introduce a new modern object-oriented system thermal-hydraulics code named RETA. RETA is characterized by several key features, including: second-order temporal and spatial discretization schemes for both incompressible and compressible flows, as well as single-phase and two-phase flows; fully-implicit solution schemes utilizing both the Newton and PJFNK methods; and an object-oriented design of major fluid components and physical models, which allows for easy extension. A series of verification and demonstration studies are conducted to verify the implementation, demonstrate the accuracy improvement, and evaluate the performance of the new code. It is concluded that results from the second-order scheme are generally more reliable. Without the concern of a code stability issue, the second-order scheme should be the preferred and default choice for a code user or an analyst.

A Conforming Virtual Element Method for Parabolic Integro-Differential Equations

Abstract This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler’s scheme for temporal discretization. With the help of Ritz–Voltera and L 2 L^{2} projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings.

Fully discrete scheme for a time-dependent Ginzburg-Landau equation in macromolecular microsphere composite hydrogels

In this article, we propose and analyze a fully discrete scheme for the time-dependent Ginzburg-Landau equation, which describes the phase field evolution of macromolecular microsphere composite hydrogels. This fully discrete scheme is a combination of the mass lumped finite element approximation for spatial discretization and the first order backward Euler scheme for temporal discretization. Besides, the convex-concave decomposition is adopted technically in the treatment of energy functional, where the concave part is treated explicitly and the convex part is treated implicitly. Moreover, we obtain the unique solvability and unconditional energy stability of the scheme. Finally, several numerical experiments are presented to illustrate the convergence order and effectiveness of the presented full-discrete scheme.

Energy-stable finite element method for a class of nonlinear fourth-order parabolic equations

In this paper, an energy-stable finite element method with the Crank–Nicolson type of temporal discretization scheme is developed and analyzed for a class of nonlinear fourth-order parabolic equations (including the Swift–Hohenberg (SH) equation and the extended Fisher–Kolmogorov (EFK) equation). In addition to the energy stability properties, the optimal spatial convergence properties in both L∞(L2)- and L∞(H1)-norm and the second-order temporal approximation rate are also obtained for numerical solutions approximating to the real solution and its Laplacian for the developed energy-stable finite element method in both semi- and fully discrete schemes. Numerical experiments are carried out to validate all attained theoretical results.

CFD Simulation of Low Speed Turbulent Flow Problems Using Unsteady Rans and Large Eddy Simulation Approach

The present paper focuses on the recent development of an implicit pressure-based finite volume algorithm for numerical solution of Navier Stokes equation in an inertial frame of reference for prediction of unsteady incompressible flow problems. The algorithm uses boundary-conforming, multiblock structured grid with moving boundaries, collocated variable arrangement with momentum equations resolved along cartesian directions, second order accurate spatial and temporal discretisation schemes for the convective fluxes and a pressurevelocity solution strategy. Effect of turbulence is simulated using one of the two different approaches. In the Unsteady Reynolds Averaged (URANS) approach coupled to appropriate eddy viscosity based turbulence models, the Navier Stokes (NS) equations, averaged over the whole range of turbulent length scales of the flow, are solved numerically. On the other hand, in the Large Eddy Simulation (LES) approach, the model filtered 3D NS equations are directly solved for the flow variables to resolve the large scale turbulent motions whereas the transport processes at the fine subgrid scale level only are simulated using simple algebraic turbulence model. The capabilities and limitations of both the cost-effective URANS approach and the relatively expensive but rich in physics LES approach have been demonstrated for few application problems of engineering interest.