Second-order Accurate Numerical Method Research Articles

Linear, Second-Order Accurate, and Energy Stable Scheme for a Ternary Cahn–Hilliard Model by Using Lagrange Multiplier Approach

We develop a second-order accurate, energy stable, and linear numerical method for a ternary Cahn–Hilliard (CH) model. The proposed scheme is an extension of typical Lagrange multiplier approach for binary CH system. The second-order backward difference formula (BDF2) is applied to construct time discretization. We theoretically prove the mass conservation, unique solvability, and energy stability of the proposed scheme. We efficiently solve the resulting discrete linear system by using a multigrid algorithm. The numerical solutions demonstrate that the proposed scheme is practically stable and second-order accurate in time and space. Moreover, we can use the proposed scheme as an effective solver to calculate the ternary CH equations in ternary phase-field fluid systems.

A variant of stabilized-scalar auxiliary variable (S-SAV) approach for a modified phase-field surfactant model

In this article, we develop a new linear, decoupled, second-order accurate, and energy stable numerical method for a modified phase-field surfactant model (Xu et al., 2020). The proposed scheme is a simple and efficient variant of stabilized-scalar auxiliary variable (S-SAV) method. The proposed scheme not only retains all advantages of S-SAV method but also simplifies the solution algorithm. The phase-field, surfactant, and auxiliary variables are totally decoupled in time, thus we can solve the whole system in a step-by-step manner. The phase-field function ϕ and surfactant ψ can be separately updated by solving two linear semi-implicit systems with constant coefficients and then the auxiliary variable is directly updated in an explicit way. We analytically prove the energy stability and the unique solvability of the proposed method. The numerical experiments show the desired temporal accuracy and energy stability. We numerically investigate the proper stabilization coefficients for the present scheme with specific parameters. Furthermore, various two- and three-dimensional benchmark tests are performed to study the dynamics of surfactant-laden phase separation.

Second-Order Accurate TVD Numerical Methods for Nonlocal Nonlinear Conservation Laws

We present a second-order accurate numerical method for a class of nonlocal nonlinear conservation laws called the "nonlocal pair-interaction model" which was recently introduced by Du, Huang, and LeFloch. Our numerical method uses second-order accurate reconstruction-based schemes for local conservation laws in conjunction with appropriate numerical integration. We show that the resulting method is total variation diminishing (TVD) and converges towards a weak solution. In fact, in contrast to local conservation laws, our second-order reconstruction-based method converges towards the unique entropy solution provided that the nonlocal interaction kernel satisfies a certain growth condition near zero. Furthermore, as the nonlocal horizon parameter in our method approaches zero we recover a well-known second-order method for local conservation laws. In addition, we answer several questions from the paper from Du, Huang, and LeFloch concerning regularity of solutions. In particular, we prove that any discontinuity present in a weak solution must be stationary and that, if the interaction kernel satisfies a certain growth condition, then weak solutions are unique. We present a series of numerical experiments in which we investigate the accuracy of our second-order scheme, demonstrate shock formation in the nonlocal pair-interaction model, and examine how the regularity of the solution depends on the choice of flux function.

A second-order numerical method for the aggregation equations

Inspired by so-called TVD limiter-based second-order schemes for hyperbolic conservation laws, we develop a formally second-order accurate numerical method for multi-dimensional aggregation equations. The method allows for simulations to be continued after the first blow-up time of the solution. In the case of symmetric, λ \lambda -convex potentials with a possible Lipschitz singularity at the origin, we prove that the method converges in the Monge–Kantorovich distance towards the unique gradient flow solution. Several numerical experiments are presented to validate the second-order convergence rate and to explore the performance of the scheme.

A second-order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel

A second-order accurate numerical method with graded meshes is proposed and analyzed for an evolution equation with a weakly singular kernel. The graded meshes are employed to compensate for the singular behavior of the exact solution at t = 0 . For the time discretization, the product integration rule is used to approximate the Riemann–Liouville fractional integral, a generalized Crank–Nicolson time-stepping is considered and shown that the error is of order k 2 , where k denotes the maximum time step. A fully discrete difference scheme is constructed with space discretization by compact difference method. Numerical experiment is carried out to support the theoretical results. The comparison between the method on uniform grids and graded grids shows the efficiency of our method.

Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface

In this paper we propose a second-order accurate numerical method to solve elliptic problems with discontinuous coefficients (with general non-homogeneous jumps in the solution and its gradient) in 2D and 3D. The method consists of a finite-difference method on a Cartesian grid in which complex geometries (boundaries and interfaces) are embedded, and is second order accurate in the solution and the gradient itself. In order to avoid the drop in accuracy caused by the discontinuity of the coefficients across the interface, two numerical values are assigned on grid points that are close to the interface: a real value, that represents the numerical solution on that grid point, and a ghost value, that represents the numerical solution extrapolated from the other side of the interface, obtained by enforcing the assigned non-homogeneous jump conditions on the solution and its flux. The method is also extended to the case of matrix coefficient. The linear system arising from the discretization is solved by an efficient multigrid approach. Unlike the 1D case, grid points are not necessarily aligned with the normal derivative and therefore suitable stencils must be chosen to discretize interface conditions in order to achieve second order accuracy in the solution and its gradient. A proper treatment of the interface conditions will allow the multigrid to attain the optimal convergence factor, comparable with the one obtained by Local Fourier Analysis for rectangular domains. The method is robust enough to handle large jump in the coefficients: order of accuracy, monotonicity of the errors and good convergence factor are maintained by the scheme.

Evolution of Star Clusters in Time-variable Tidal Fields

Abstract Strong tidal forces can dominate star cluster evolution in merging galaxies, determining their mass-loss rates and lifetimes. In order to model this evolution, we have developed a second-order accurate numerical method for integrating a star cluster in an arbitrary time-variable tidal field. We extend the Kira N-body integrator to handle these external fields. We obtain realistic tidal histories from a galaxy merger simulation including sink particles, which we interpret as young star clusters. Coupling these tidal accelerations to N-body models of isolated clusters, we perform detailed dynamical studies. This generalizes the formalism previously used to explore the dynamical effects of the galactic tidal field on clusters in circular orbits. We find that, in contrast to previous studies that considered only stellar and dark matter dynamics, tidal interactions between clusters and dense gas in the galactic disk can significantly influence cluster mass loss and lifetimes. Using our models, we develop an effective semianalytic model that can be used for fast estimation of cluster mass loss in a galactic tidal field and to study the evolution of the globular cluster mass function in isolated and merging galaxies.

A second-order accurate numerical method for the space–time tempered fractional diffusion-wave equation

This paper focuses on providing the high order algorithms for the space–time tempered fractional diffusion-wave equation. The designed schemes are unconditionally stable and have the global truncation error O ( τ 2 + h 2 ) , being theoretically proved and numerically verified.

Eulerian--Lagrangian Treatment of Nondilute Two-Phase Gels

We develop a three-dimensional Eulerian--Lagrangian two phase mixture model for a gel in which fluid is represented in Eulerian coordinates and network in Lagrangian coordinates. The equations of motion and boundary conditions in their respective frames are derived using a principle of virtual power. The volume fractions of each material at each point of Eulerian space are calculated using the position and volume density of network particles. Network and fluid have separate velocity fields, and the model includes viscous, elastic, and applied forces as well as those due to chemical potentials, pressure, and intermaterial drag. We also develop a second-order accurate numerical method for solving the model equations in one dimension. Illustrations of model behavior under a variety of conditions are shown.

Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations

We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.

Two-Dimensional Numerical Simulation of Gas–Solid Reactive Flow with Moving Boundary

In order to combine the advantages of both the Lagrangian and Eulerian algorithm for a moving boundary, this work presents a two-dimensional axisymmetric computational model in ALE (arbitrary Lagrangian–Eulerian) forms for the gas–solid transient reacting flow with a moving boundary of the interior ballistic process. A two-phase flow model is established to describe the complex physical process based on a modified two-fluid theory, which takes into account gas production, interphase drag, intergranular stress, and heat transfer between two phases. The governing equations are discretized with the TVD-type MUSCL scheme to obtain a second-order accurate numerical method in finite volume form and solved by the semi-implicit method for pressure-linked equations with density corrections. A dynamic self-adapting mesh update method is developed to expand the computational domain for projectile motion and reduce the computational cost. Several verification tests demonstrate the accuracy and reliability of this approach. A pressure-driven projectile case is used to demonstrate the coupling of the moving projectile with gas dynamics. The application on a real gun shows an excellent agreement between numerical simulation and experimental measurements. Numerical results provide a deeper understanding of the interior ballistic process, including gas production, flame spreading, and pressure wave developing, etc. By applying the ALE technique to two-phase reactive flows with moving boundary, it is be able to take advantage of the best aspects of both Lagrangian and Eulerian approaches. This new method is reliable as a predictive tool for the study of the physical phenomenon and can therefore be used as an assessment tool for future interior ballistics studies.

Numerical Method for Solving Matrix Coefficient Elliptic Equation on Irregular Domains with Sharp-Edged Boundaries

We present a new second-order accurate numerical method for solving matrix coefficient elliptic equation on irregular domains with sharp-edged boundaries. Nontraditional finite element method with non-body-fitting grids is implemented on a fictitious domain in which the irregular domains are embedded. First we set the function and coefficient in the fictitious part, and the nonsmooth boundary is then treated as an interface. The emphasis is on the construction of jump conditions on the interface; a special position for the ghost point is chosen so that the method is more accurate. The test function basis is chosen to be the standard finite element basis independent of the interface, and the solution basis is chosen to be piecewise linear satisfying the jump conditions across the interface. This is an efficient method for dealing with elliptic equations in irregular domains with non-smooth boundaries, and it is able to treat the general case of matrix coefficient. The complexity and computational expense in mesh generation is highly decreased, especially for moving boundaries, while robustness, efficiency, and accuracy are promised. Extensive numerical experiments indicate that this method is second-order accurate in the L∞ norm in both two and three dimensions and numerically very stable.

A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation

In recent years, it has been found that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from Fractional Calculus. Recently, a new space and time fractional Bloch-Torrey equation (ST-FBTE) has been proposed (Magin et al., J. Magn. Reson. 190(2), 255---270, 2008), and successfully applied to analyse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the ST-FBTE with a nonlinear source term on a finite domain in three-dimensions. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we propose a spatially second-order accurate implicit numerical method (INM) for the ST-FBTE whereby we discretize the Riesz fractional derivative using a fractional centered difference. Secondly, we prove that the implicit numerical method for the ST-FBTE is uniquely solvable, unconditionally stable and convergent, and the order of convergence of the implicit numerical method is O ? 2 ? ? + ? + h x 2 + h y 2 + h z 2 $O\left (\tau ^{2-\alpha }+\tau +h_{x}^{2}+h_{y}^{2}+h_{z}^{2}\right )$ . Finally, some numerical results are presented to support our theoretical analysis.

Implicit multiblock method for solving a kinetic equation on unstructured meshes

A parallel multiblock implementation of a second-order accurate implicit numerical method based on solving a model kinetic equation is proposed for analyzing three-dimensional rarefied gas flows. The performance of the method is illustrated by computing test examples of gas flows in a circular pipe in a wide range of Knudsen numbers. The convergence rate and scalability of the method are analyzed depending on the number of blocks in the spatial grid.

Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

AbstractRecently, the fractional Bloch-Torrey model has been used to study anomalous diffusion in the human brain. In this paper, we consider three types of space and time fractional Bloch-Torrey equations in two dimensions: Model-1 with the Riesz fractional derivative; Model-2 with the one-dimensional fractional Laplacian operator; and Model-3 with the two-dimensional fractional Laplacian operator.Firstly, we propose a spatially second-order accurate implicit numerical method for Model-1 whereby we discretize the Riesz fractional derivative using a fractional centered difference. We consider a finite domain where the time and space derivatives are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Secondly, we utilize the matrix transfer technique for solving Model-2 and Model-3. Finally, some numerical results are given to show the behaviours of these three models especially on varying domain sizes with zero Dirichlet boundary conditions.

Inverse problem on the semi-axis: local approach

We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.

Inverse problem on the semi-axis: local approach

We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.

Dynamical inverse problem on a metric tree

We consider the problem of reconstruction of the potential for the wave equation on a specific star graph using the dynamical Dirichlet-to-Neumann map. Our algorithm is based on the boundary control method. We reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples demonstrate that the algorithm works and can be used to reconstruct an unknown potential accurately.

A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel

We study a generalized extrapolated Crank-Nicolson scheme for the time discretization of a semilinear integro-differential equation with a weakly singular kernel, in combination with a space discretization by linear finite elements. The scheme uses variable grids in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k 2 + h 2 , where k and h are the parameters for the time and space meshes, respectively. The results of numerical computations demonstrate the convergence of our scheme.

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k2 + h2, where k and h are the parameters for the time and space meshes, respectively.