Second-order Scheme Research Articles

An equidistributed grid-based second-order scheme for a singularly perturbed Fredholm integro-differential equation with an interior layer

In this study, we introduce and evaluate the effectiveness of the central finite difference coupled with a composite trapezoidal rule on adapted meshes in solving a second-order singularly perturbed Fredholm integro-differential equation with a discontinuous source term. A grid equidistribution technique was employed to address the challenge posed by the perturbation parameter in classical numerical methods. Our approach demonstrated superior precision compared to existing schemes in the literature and achieved second-order uniform convergence.

Error analysis of second-order IEQ numerical schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation

In this article, we derive error estimates for two second-order numerical schemes for solving the viscous Cahn-Hilliard equation with hyperbolic relaxation, one based on the second-order Crank-Nicolson time marching method and the other on the backward differentiation formula. In both schemes, the nonlinear potential is discretized by the Invariant Energy Quadratization (IEQ) method, which employs an auxiliary variable to produce a linear and unconditional energy-stable structure. After assuming some reasonable boundedness and continuity conditions for the nonlinear function, the optimal error estimate is rigorously derived using mathematical induction. Finally, several numerical experiments are carried out to verify the theoretical predictions of the convergence rate and energy stability of the algorithms.

Analysis of a second-order numerical scheme for time-fractional partial integro-differential equations with a weakly singular kernel

In this paper, a second-order numerical scheme is developed and analyzed to approximate the solution of a class of time-fractional Volterra integro-differential equations with a weakly singular kernel on a rectangular domain. In the present scheme, we first discretize the domain on a uniform grid. The L1-2 technique is used to discretize the Caputo derivative of order α with α∈(0,1) in the temporal direction. A weakly singular kernel makes it more challenging to approximate the integral and reduces the accuracy of the classical numerical integration method. In order to improve the convergence rate, the composite product trapezoidal formula is employed to approximate the fractional integral. Secondly, a classical central difference technique is used for spatial discretization. Using a maximum norm, the stability and convergence analysis is carried out under suitable regularity assumptions. It is proved that the scheme achieves second-order accuracy. In addition, we extend the proposed difference scheme to solve the corresponding semilinear problem. The well-known Newton’s linearization technique is used to deal with semilinearity. Numerical examples demonstrate the efficiency and applicability of the proposed scheme.

A continuous one-domain framework for fluid flow in superposed clear and porous media

Based on generalized porous flow model, we present a one-domain framework for the simulation of incompressible flow in superposed clear and porous media. The continuous change in the characteristics of the porous medium is naturally taken into account by the model. In presence of any strong discontinuities in the domain, the sudden changes are approximated as a continuous but rapid variations over a thin layer across the regions of contrast. The governing equations are solved by a consistent second-order projection scheme on finite volume octree cells. The numerical scheme evaluates advection as well as porous drag terms at fractional time step and employs Godunov type procedure to obtain fractional step velocity at face center. The consistent property of the scheme ensures a discrete balance between pressure gradient and porous drags and, thereby, recovers Darcy-Forchheimer law exactly. The implication of inconsistent discretization of porous drags and pressure gradient is also highlighted. Benchmark problems with flow parallel and perpendicular to the homogeneous porous layer are considered to test the accuracy of the proposed numerical model especially with regard to the artificial transition region. Although the incorporation of the transition layer introduces error into the system, it reduces non-linearity in the field variables and, therefore, decreases the demand of higher grid density near the interface in one-domain approach. Furthermore, we simulate a series of physical problems with varying complexity in flow and characteristics of porous medium. The simulation results illustrate the ability of our solver to handle flows with viscous as well as inertial effects around a planar or circular interface geometry without loss of stability.

Efficient interior penalty discontinuous Galerkin projection method with unconditional energy stability and second-order temporal accuracy for the incompressible magneto-hydrodynamic system

In this paper, we propose a novel interior penalty discontinuous Galerkin projection method for the incompressible magneto-hydrodynamic equations. The scheme is employed by an implicit-explicit treatment of the nonlinear coupling terms and a second-order rotational pressure-correction scheme for dealing with the Navier-Stokes equations. One noteworthy aspect of this scheme is the introduction of an additional stabilization term to Maxwell's equations, which allows for the explicit treatment of the coupled nonlinear terms and the decoupling of computations for the magnetic and velocity fields, ultimately leading to the achievement of desired linearity, full decoupling, second-order accuracy in time, and unconditional energy stability. The obtained scheme is easy to implement because one only needs to solve a few decoupled linear equations at each time step. We rigorously prove the unique solvability and unconditional energy stability of the developed scheme and present a series of numerical examples to demonstrate the accuracy, stability, and efficiency of the proposed scheme.

A new open-source library based on novel high-resolution structure-preserving convection schemes

This work develops a new finite volume convection scheme and open-source library for convection-dominated problems on unstructured grids. The formulated reconstruction operators can realize high-resolution and structure-preserving properties by adjusting numerical dissipation and anti-dissipation errors in a unified normalized-variable diagram. Then, this work develops a new open-source library based on the devised schemes within one of the most widely used general-purpose open-source software whose solution quality is intrinsically limited by the low accuracy of the conventional total variation diminishing schemes. The performance of the new library is first verified by some fundamental problems including pure advection and inviscid Euler equations. The numerical results of the accuracy tests show that the developed interpolation library can significantly reduce numerical errors with a minor increased computational cost compared with the conventional second-order schemes. The complex profile advection test demonstrates that the new library can offer an improved structure-preserving property that gives essentially oscillation-free solutions and preserves the structures of the passive transported scalar. Furthermore, the robustness of the new library is validated by compressible high-speed flow simulations with irregular geometries. Thus, this work presents a new library of high-resolution structure-preserving schemes for the accurate and efficient discretization of the convection term which plays an essential role in a wide range of computational fluid dynamics. The source code of the new scheme and library are released via DOI:10.5281/zenodo.8399000.

Stability, convergence and error analysis of B-spline collocation with Crank–Nicolson method and finite element methods for numerical solution of Schrodinger equation arises in quantum mechanics

In the present study, the complex-valued Schrodinger equation (CVSE) is solved numerically by a nonic B-spline finite element method (FEM) in quantum mechanics. The approach employed is based on the collocation of nonic B-splines over spatial finite elements, so that we have continuity of the dependent variable and its first eight derivatives throughout the solution range. For time discretization, the Crank-Nicolson scheme of second order based on FEM is employed. The method is shown to be unconditionally stable and accurate to order. Three problems are considered to validate the algorithm. Comparisons are made with existing methods and analytical solutions. The proposed method exhibits good conservation properties and performs well with regards to analytical solutions for different error norms and conservative constants related to parameters in quantum classes in Physics. The computational complexity of (2N+18) arithmetic operations with the help of a nonic-diagonal matrix is also tackled by the present scheme.

Fast parallel IGA-ADS solver for time-dependent Maxwell's equations

We propose a simulator for time-dependent Maxwell's equations with linear computational cost. We employ B-spline basis functions as considered in the isogeometric analysis (IGA). We focus on non-stationary Maxwell's equations defined on a regular patch of elements. We employ the idea of alternating-directions splitting (ADS) and employ a second-order accurate time-integration scheme for the time-dependent Maxwell's equations in a weak form. After discretization, the resulting stiffness matrix exhibits a Kronecker product structure. Thus, it enables linear computational cost LU factorization. Additionally, we derive a formulation for absorbing boundary conditions (ABCs) suitable for direction splitting. We perform numerical simulations of the scattering problem (traveling pulse wave) to verify the ABC. We simulate the radiation of electromagnetic (EM) waves from the dipole antenna. We verify the order of the time integration scheme using a manufactured solution problem. We then simulate magnetotelluric measurements. Our simulator is implemented in a shared memory parallel machine, with the GALOIS library supporting the parallelization. We illustrate the parallel efficiency with strong and weak scalability tests corresponding to non-stationary Maxwell simulations.

Refined Schemes for Computing the Dynamics of Elastoviscoplastic Media

For a stable numerical solution of the system of equations governing an elastoviscoplastic continuous medium model, a second-order explicit-implicit scheme is proposed. An explicit approximation is used for the equations of motion, and an implicit approximation, for the constitutive relations containing a small relaxation time parameter in the denominator of the nonlinear free terms. A second-order implicit approximation for isotropic and anisotropic elastoviscoplastic models is constructed to match the orders of approximation of the explicit elastic and implicit adjustment steps. Refined formulas for correcting the stress deviators after the elastic step are derived for various viscosity function representations. The resulting solutions of the second-order implicit approximation for the stress deviators of the elastoviscoplastic equations admit passage to the limit as the relaxation time tends to zero. The correcting formulas obtained via this passage to the limit can be treated as regularizers of the numerical solutions to the elastoplastic systems.

A second-order scheme based on blended BDF for the incompressible MHD system

A second-order scheme based on blended BDF for the incompressible MHD system

A second-order fully-balanced structure-preserving variational discretization scheme for the Cahn–Hilliard–Navier–Stokes system

We propose and analyze a structure-preserving space-time variational discretization method for the Cahn–Hilliard–Navier–Stokes system. Uniqueness and stability for the discrete problem is established in the presence of concentration-dependent mobility and viscosity parameters by means of the relative energy estimates and order optimal convergence rates are established for all variables using balanced approximation spaces and relaxed regularity conditions on the solution. Numerical tests are presented to demonstrate the proposed method is fully practical and yields the predicted convergence rates. The discrete stability estimates developed in this paper may also be used to analyse other discretization schemes, which is briefly outlined in the discussion.

Investigation of the effects of volume change on flow structure and acoustic in a silencer

This study numerically examined the propellant flow from gunfire using the kω-SST turbulence model and their sound attenuation using the Ffowcs-Williams and Hawkins equations (FW-H). For simulation, a pressure-based solver and 3D axisymmetric geometry were used. The second-order implicit time approach and the second-order upwind scheme spatial discretization were used in the simulation. The maximum exit pressure was 3.748 MPa for the suppressor with a length of 70 mm and diameter of 20 mm. However, when the diameter suppressor increased by 1/6, the maximum exit pressure was reduced to 3.4961 MPa. When the length increased by 1/6, the maximum pressure became 3.3636 MPa. Lastly, when the diameter and length were increased by 1/6, the maximum exit pressure became 3.177 MPa. For this suppressor, 20.835 dB (12.29%) sound pressure level attenuation was achieved with 16.823 MPa (84.115%) overpressure reduction and 484.86 K or 32.32% temperature reduction. Generally, the attenuation increased with the increase in the suppressor’s internal volume.

A novel estimation method for microstructural evolution based on data assimilation and phase field crystal model

In the simulation of microstructural evolutions, detailed priori knowledge for the model parameters and initial states is difficult to be observed by experiments. For an improved simulation, we present a data assimilation framework for the phase field crystal model with the effect of stochastic noise. A sequential data assimilation method based on the ensemble Kalman filter is applied to integrate phase field simulation and observational data from experiments. Furthermore, we couple the present framework with a second-order accurate unconditional-energy stable scheme, and thus the issue of stability restriction can be avoided. A series of twin experiments are performed to demonstrate the performance of our present framework. The results reveal that our framework can successfully improve the simulation accuracy, even when some parameters are uncertain and the initial value significantly deviates from the observed situation.

Linearly implicit energy-preserving integrating factor methods and convergence analysis for the 2D nonlinear Schrödinger equation with wave operator

Abstract In this paper, we develop a novel class of linearly implicit and energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. To begin, a second-order scheme is proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to circumvent the difficulty arising from the unavailability of a priori$L^{\infty }$ estimates of numerical solutions. Based on the integrating factor Runge–Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linearly implicit and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.

Natural convection from solid and hollow cylinders with concave surface: A numerical approach

The buoyancy-driven thermofluid dynamics of a solid/hollow cylinder with concave surfaces have been studied in a laminar regime () using the finite-volume method to analyze the effects of ( and . Conservation equations for mass, momentum, and energy have also been solved in a 2-D axisymmetric geometry . Second-order upwind and central difference schemes have been deployed to discretize the convective and diffusive terms. Heat dissipation increases with irrespective of for both cylinders. However, the remains constant with the ratio for a hollow concave cylinder increases with whereas, decreases. Multiple vortices appear at the outside surface at a low (i.e. = 0.2), and these vortices almost disappear at high (i.e. = 0.6), affecting the outside average Nusselt number from low to high. In all cases, is higher than The thermofluid dynamics of solid and hollow cylinders have been delineated through different countour and vector plots, and two correlations for Nu have been deveoped.

Two fast and unconditionally stable finite difference methods for Riesz fractional diffusion equations with variable coefficients

In this paper, for variable coefficient Riesz fractional diffusion equations in one and two dimensions, we first design a second-order implicit difference scheme by using the Crank-Nicolson method and a fractional centered difference formula for time and space variables, respectively. With the compact operator acting on, a novel fourth-order finite difference scheme is subsequently constructed. Solvability, stability and convergence of these schemes are theoretically analyzed. For these discretized linear systems, the fast implementation with preconditioners based on sine transform is proposed, which has the computational complexity of O(nlog⁡n) per iteration and the memory requirement of O(n), where n represents the total number of the spatial grid nodes. Finally, numerical experiments are performed to illustrate the preciseness and effectiveness of these new techniques.

An efficient HLL-based scheme for capturing contact-discontinuity in scalar transport by shallow water flow

An efficient numerical scheme is developed for coupled hydrodynamic and scalar transport systems to guarantee the conservation and the positivity-preserving properties for water depth and scalar concentration. A second-order well-balanced positivity-preserving central-upwind scheme based on the finite volume method is adopted to discretize both the Saint-Venant system and the advective fluxes in the scalar transport system. In particular, an anti-diffusion modification is augmented in an ad hoc manner to the Harten–Lax–van Leer approximate Riemann solver for the scalar transport system, with the aim of significantly reducing the numerical diffusion near contact discontinuities. The proposed scheme is validated through seven numerical experiments, wherein the advection and diffusion processes of scalar transport are considered either separately or in combination. Accuracy is measured based on the difference between the numerical results and analytical solutions, and the performance of the proposed scheme is assessed by comparing it with that of the existing scheme. Convergence analysis confirms that the proposed scheme is accurate to the second order. The stationary steady state with wet-dry fronts is simulated, which verifies that the proposed scheme can preserve the exact C-property. The results for the pure diffusion case reveal that the original scheme has limitations in precisely predicting scalar diffusion when the diffusion coefficient is small. By contrast, the proposed scheme accurately approximates scalar diffusion, even at low diffusivity. The results under flow conditions with various Froude and Péclet numbers confirm that the proposed scheme is more accurate than the existing scheme in solving scalar transport problems over a wide range of shallow water flows.

Theoretical Study of Non-Isothermal Gradient Elution Liquid Chromatography.

A two-component model of gradient elution chromatography is investigated to theoretically study the effects of simultaneous variations in temperature and solvent strength on the retention behaviors of elution profiles in thermally insulated liquid chromatographic columns. The gradient elution technique is based on the gradual increase or decrease in eluent strength during the chromatographic operation by varying the composition of the mobile phase. The enthalpy of adsorption is primarily responsible for internal temperature variations inside the column, as heat adsorbs during the adsorption process and releases in the desorption phase. Both types of variations change the propagation speeds of moving pulses inside the column which can lead to better separation of the components and a reduction in the recycling time for the next injection. The equilibrium dispersive model (EDM) coupled with the energy balance equation for temperature and transport equation for the volume fraction of the solvent is utilized to simulate this complex process. The resulting nonlinear model equations are approximated by applying a semi-discrete second-order finite volume scheme. The numerical solutions are used to study the impact of a gradient starting and ending times, volume-fraction of the solvent, solvent strength parameter, the slope of gradient, enthalpy of adsorption, injection temperature, and the ratio of specific heats on the concentration profiles.

Phase-field simulation of multiple fluid vesicles with a consistently energy-stable implicit–explicit method

When certain polymer amphiphiles and phospholipids are dispersed in a liquid, these molecules combine to form various closed bilayer structures known as multiple vesicles. The specific structures of these vesicles play fundamental roles in cytobiology and drug transportation. To model the multiple lipid vesicles in a fluid environment, we use the multi-phase conservative Allen–Cahn-type equations with approximately area-preserving penalty terms to capture the vesicle interfaces and the incompressible Navier–Stokes equations to describe the evolution of velocity and pressure. To efficiently simulate this complex coupled fluid system, we introduce several time-dependent variables to transform the total energy functional and the governing equations into equivalent forms. Based on these equivalent models, we develop a totally decoupled, energy-stable, and temporally second-order accurate time-marching scheme. A consistency-enhanced technique is utilized to correct the computed energy so that a more accurate solution is obtained. The time-discretized energy stability can be analytically estimated and the solution algorithm in each time step is easy to implement. Various numerical experiments, such as multi-component vesicles with different initial shapes, multiple vesicles in shear flow, and multiple vesicles in complex domains, are performed to verify the accuracy, stability, and capability of our proposed method.

Efficient Finite Difference Approaches for Solving Initial Boundary Value Problems in Helmholtz Partial Differential Equations

This study presents numerical solutions for initial boundary value problems of homogeneous and nonhomogeneous Helmholtz equations using first- and second-order difference schemes. The stability of these methods is rigorously analyzed, ensuring their reliability and convergence for a wide range of problem instances. The proposed schemes’ robustness and applicability are demonstrated through several examples, accompanied by an error analysis table and illustrative graphs that visually represent the accuracy of the solutions obtained. The results confirm the effectiveness and efficiency of the proposed schemes, making them valuable tools for solving Helmholtz equations in practical applications.