Second-order Convergence Research Articles

Optimal time two-mesh mixed finite element method for a nonlinear fractional hyperbolic wave model

<abstract><p>In this article, a second-order time discrete algorithm with a shifted parameter $ \theta $ combined with a fast time two-mesh (TT-M) mixed finite element (MFE) scheme was considered to look for the numerical solution of the nonlinear fractional hyperbolic wave model. The second-order backward difference formula including a shifted parameter $ \theta $ (BDF2-$ \theta $) with the weighted and shifted Grünwald difference (WSGD) approximation for fractional derivative was used to discretize time direction at time $ t_{n-\theta} $, the $ H^1 $-Galerkin MFE method was applied to approximate the spatial direction, and the fast TT-M method was used to save computing time of the developed MFE system. A priori error estimates for the fully discrete TT-M MFE system were analyzed and proved in detail, where the second-order space-time convergence rate in both $ L^2 $-norm and $ H^1 $-norm were derived. Detailed numerical algorithms with smooth and weakly regular solutions were provided. Finally, some numerical examples were provided to illustrate the feasibility and effectiveness for our scheme.</p></abstract>

Computational analysis of time fractional diffusion wave equation via novel B-splines-based technique

<abstract><p>In this paper, the authors propose and utilize a novel B-splines-based technique to solve time fractional diffusion wave equations numerically. The present approach employs the arbitrary-order fractional derivative in conjunction with the $ \theta $-weighted scheme. The integration of the Atangana-Baleanu fractional derivative into time fractional diffusion wave equations represents a novel advancement within B-spline methodologies. Furthermore, the stability and convergence of the presented schemes are given. The suggested approach not only exhibits unconditional stability, it also attains second-order convergence in both temporal and spatial dimensions. We conclude by applying our theoretical results to certain numerical examples to demonstrate their efficiency and accuracy.</p></abstract>

A numerical approach for solving Volterra delay integro-differential equation

In this paper, we propose a finite difference technique for approximating solutions of first-order Volterra delay integro-differential equation. The presented numerical method acquires a second-order convergence in discrete maximum norm. The derived results are numerically validated in test problems to support the theoretical analysis.

Dynamic analysis of wave scenarios based on enhanced numerical models for the good Boussinesq equation

The good Boussinesq equation, a modification of the Boussinesq equation, aims to enhance predictions about shallow water wave behavior. This paper introduces two finite difference schemes for solving the good Boussinesq equation including linear and nonlinear implicit finite difference methods. Both schemes utilize the pseudo-compact difference approach, delivering second-order precision with an additional term to boost numerical simulation accuracy while maintaining the grid points of the standard scheme. These schemes rigorously preserve the critical physical characteristics of the good Boussinesq equation, ensuring more precise representation. We establish the existence of solutions with discrete differences and demonstrate, through the discrete energy method, their uniqueness, stability, and second-order convergence in the maximum norm. Furthermore, we propose an iterative algorithm tailored for the nonlinear implicit finite difference scheme, resulting in significant reductions in computational costs compared to the linear scheme. The results of our numerical experiments demonstrate that our methods are competitive and efficient when compared to difference schemes and previously used methods, while maintaining crucial physical qualities. Furthermore, we run relevant numerical simulations to demonstrate the accuracy of the current methods using evidence from the solitary wave interaction with the initial amplitudes of the wave. It is also suggested that the issue has a critical initial wave amplitude for the interaction of two solitary waves, where blow up occurs in a finite amount of time for initial wave amplitudes greater than the new blow-up criteria value.

Non-ideal magnetohydrodynamics on a moving mesh II: Hall effect

ABSTRACT In this work, we extend the non-ideal magnetohydrodynamics (MHD) solver in the moving-mesh code arepo to include the Hall effect. The core of our algorithm is based on an estimation of the magnetic field gradients by a least-squares reconstruction on the unstructured mesh, which we also used in the companion paper for Ohmic and ambipolar diffusion. In an extensive study of simulations of a magnetic shock, we show that without additional magnetic diffusion our algorithm for the Hall effect becomes unstable at high resolution. We can however stabilize it by artificially increasing the Ohmic resistivity, ηOR, so that it satisfies the condition ηOR ≥ ηH/5, where ηH is the Hall diffusion coefficient. Adopting this solution, we find second-order convergence for the C-shock and are also able to accurately reproduce the dispersion relation of the whistler waves. As a first application of the new scheme, we simulate the collapse of a magnetized cloud with constant Hall parameter ηH and show that, depending on the sign of ηH, the magnetic braking can either be weakened or strengthened by the Hall effect. The quasi-Lagrangian nature of the moving-mesh method used here automatically increases the resolution in the forming core, making it well suited for more realistic studies with non-constant magnetic diffusivities in the future.

Structure-preserving discretization of fractional vector calculus using discrete exterior calculus

Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties curlαgradα=0 and divαcurlα=0 hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size.

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for vorticity-streamfunction formulation

In this work, we propose a non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) for the two-dimensional (2D) incompressible Navier-Stokes equations in vorticity-stream function formulation. In this method, the vorticity and the streamfunction equations are calculated by two separate LB equations, which are constructed by using the non-orthogonal transformation matrix. Most importantly, to eliminate the residual term that may appear in the recovered vorticity equation, the convection term is handled as a source term in the LB equation rather than being directly recovered by the evolution equation. In addition, due to the convection term in our method is computed with a local scheme, it retains the parallel feature of the standard LBM. We then test the accuracy and the stability of the present LBM by considering various benchmark examples. The results show that the present method has a second-order convergence rate in space. Our numerical results indicate that the present LBM is a good candidate for solving streamfunction-vorticity formulation.

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.

A monotone finite volume scheme for single phase flow with reactive transport in anisotropic porous media

This paper deals with development and analysis of a finite volume scheme for a nonlinear parabolic equation modeling a single phase flow with reactive solute transport, including equilibrium sorption, in heterogeneous porous media. We present a second-order, finite volume scheme by combining a nonlinear two point flux approximation for the diffusion term, and an upwind method for the advective term on unstructured meshes. It is shown that, under a CFL condition, this scheme ensures the non-negativity of the discrete solution, taking into account the anisotropy and heterogeneity of the domain. The existence of a solution for the discrete nonlinear system is studied, and the convergence of a fixed point method is proved. Our theoretical results are illustrated by numerical simulations in a bidimensional domain, in particular we show the second-order convergence rate for the numerical solution and confirm the monotonicity of the scheme.

An FFT-Based MAC scheme for Stokes equations with periodic boundary conditions and its application to elasticity problems

This paper presents a fast Fourier transform (FFT) based finite difference MAC scheme to compute the Stokes equations with periodic boundary conditions. The numerical implementation of this solver is very convenient and only requires three simple steps. The remarkable superiorities of such a solver compared to classical iterative approaches are computational time-saving and memory storage saving. The application of the proposed solver to nearly incompressible linear elasticity problems is also introduced. In addition, following the similar spirit used in Dong et al. (2020) for Stokes equations with Dirichlet boundary conditions, a rigorous error analysis is provided for Stokes equations with periodic boundary conditions. The results include: a second-order convergence in the ℓ2-norm for velocity, pressure as well as the gradient of velocity; a second-order accuracy in maximum norm for both velocity and its gradient. Moreover, the properties of the discrete Green functions play an important part in maximum error estimation. Most notably, the major difficulties of the extension from Dirichlet boundary conditions to periodic boundary conditions are the discrete version of Green’s formulae, Poincaré inequality and LBB condition. A variety of two-dimensional and three-dimensional numerical examples are given to illustrate the efficiency of the proposed FFT solver and validate the theoretical results.

Optimal implicit single-step time integration methods with equivalence to the second-order-type linear multistep methods for structural dynamics: Accuracy analysis based on an analytical framework

This paper presents a comprehensive study of the second-order-type time integration methods family encompassing the linear multistep (LMS) methods, the single-step methods, and the LMS equivalent single-step methods for structural dynamics. An analytical accuracy framework for algorithm design, convergence, and optimization is developed, where the obstacles encountered in algorithms construction, convergence accuracy, algorithms equivalence, accuracy measurement, and accuracy optimization in traditional accuracy frameworks are well addressed. Rigorous accuracy analysis successively to the second-order-type LMS (LMS2) methods, the single-step methods, and the LMS2 equivalent single-step methods aims to reveal optimal algorithms with desirable numerical properties. First, independent implicit algorithms including the velocity type and the acceleration type LMS2 methods except the displacement type ones are newly revealed in the LMS2 methods family with second-order convergence and A-stability, and three error constants that can play essential roles in the accuracy measurement and algorithm optimization are developed. Second, a single-step methods family where the displacement, velocity, and acceleration can achieve the same order of accuracy simultaneously is revealed in a generalized 16-parameters methods family, which contains numerous conventional single-step single-solve methods as subfamily methods. It is found that the unit consistent and non-consistent representations show a significant difference in the local truncation error, stability, and convergence performances. Third, a family of accuracy-improved single-step methods with algorithmic equivalence, which is stricter than the traditional spectral equivalence, to the LMS2 methods is revealed. It is clarified that the difference between the methods with equivalence can significantly influence their convergence accuracy, and the error constants concept of the LMS2 methods can be extended to the single-step method with algorithmic equivalence. Last, an algorithm optimization strategy is developed based on the error constants and the ultimate spectral radius, and an optimal single-step method with a single parameter to control the accuracy and numerical dissipation is revealed analytically without empirical assumptions. The optimal method is second-order accurate, unconditionally stable, overshooting acceptable, and equivalent to an optimal route between the Houbolt method and the trapezoidal rule, and its numerical performance is verified by the spectral accuracy and two numerical examples.

Splitting schemes for the semi-linear wave equation with dynamic boundary conditions

This paper introduces novel bulk–surface splitting schemes of first and second order for the wave equation with kinetic and acoustic boundary conditions of semi-linear type. For kinetic boundary conditions, we propose a reinterpretation of the system equations as a coupled system. This means that the bulk and surface dynamics are modeled separately and connected through a coupling constraint. This allows the implementation of splitting schemes, which show first-order convergence in numerical experiments. On the other hand, acoustic boundary conditions naturally separate bulk and surface dynamics. Here, Lie and Strang splitting schemes reach first- and second-order convergence, respectively, as we reveal numerically.

Recursive solution of initial value problems with temporal discretization

We construct a continuous domain, as a model of interval analysis, for temporal discretization of differential equations. By using this domain, and the domain of Lipschitz maps, we formulate a generalization of the Euler operator, which exhibits second-order convergence. We prove computability of the operator within the framework of effectively given domains. The operator only requires the vector field of the differential equation to be Lipschitz continuous, in contrast to the related operators in the literature which require the vector field to be at least continuously differentiable. Within the same framework, we also analyze temporal discretization and computability of another variant of the Euler operator formulated according to Runge-Kutta theory. We prove that, compared with this variant, the second-order operator that we formulate directly, not only imposes weaker assumptions on the vector field, but also exhibits superior convergence rate. We implement the first-order, second-order, and Runge-Kutta Euler operators using arbitrary-precision interval arithmetic, and report on some experiments. The experiments confirm our theoretical results. In particular, we observe the superior convergence rate of our second-order operator compared with the Runge-Kutta Euler and the common (first-order) Euler operators.

A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column

Abstract. A common task in oceanography is to model the vertical movement of particles such as microplastics, nanoparticles, mineral particles, gas bubbles, oil droplets, fish eggs, plankton, or algae. In some cases, the distribution of the vertical rise or settling velocities of the particles in question can span a wide range, covering several orders of magnitude, often due to a broad particle size distribution or differences in density. This requires numerical methods that are able to adequately resolve a wide and possibly multi-modal velocity distribution. Lagrangian particle methods are commonly used for these applications. A strength of such methods is that each particle can have its own rise or settling speed, which makes it easy to achieve a good representation of a continuous distribution of speeds. An alternative approach is to use Eulerian methods, where the partial differential equations describing the transport problem are solved directly with numerical methods. In Eulerian methods, different rise or settling speeds must be represented as discrete classes, and in practice, only a limited number of classes can be included. Here, we consider three different examples of applications for a water column model: positively buoyant fish eggs, a mixture of positively and negatively buoyant microplastics, and positively buoyant oil droplets being entrained by waves. For each of the three cases, we formulate a model for the vertical transport based on the advection–diffusion equation with suitable boundary conditions and, in one case, a reaction term. We give a detailed description of an Eulerian and a Lagrangian implementation of these models, and we demonstrate that they give equivalent results for selected example cases. We also pay special attention to the convergence of the model results with an increasing number of classes in the Eulerian scheme and with the number of particles in the Lagrangian scheme. For the Lagrangian scheme, we see the 1/Np convergence, as expected for a Monte Carlo method, while for the Eulerian implementation, we see a second-order (1/Nk2) convergence with the number of classes.

A Cubic B-Spline Collocation Method for Barrier Options under the CEV Model

In this paper, we construct a new numerical algorithm for the partial differential equation of up-and-out put barrier options under the CEV model. In this method, we use the Crank-Nicolson scheme to discrete temporal variables and the cubic B-spline collocation method to discrete spatial variables. The method is stable and has second-order convergence for both time and space variables. The convergence analysis of the proposed method is discussed in detail. Finally, numerical examples verify the stability and accuracy of the method.

Conservative second-order finite difference method for Camassa–Holm equation with periodic boundary condition

In this paper, we propose two momentum-preserving finite difference schemes for solving one-dimensional Camassa–Holm equation with periodic boundary conditions. A two-level nonlinear difference scheme and a three-level linearized difference scheme are constructed by using the method of order reduction. For nonlinear scheme, we combine mid-point rule and a specific difference operator, which ensures that our obtained scheme is of second-order convergence in both temporal and spatial directions. For linearized scheme, we apply a linear implicit Crank–Nicolson scheme in the temporal direction, then unique solvability and momentum conservation are analysed in detail. Numerical experiments are provided for Camassa–Holm equation admitting different types of solutions, which demonstrate the convergence order and accuracy of the proposed methods coincide with theoretical analysis. Moreover, numerical results show that the nonlinear scheme exhibits better accuracy for mass conservation, while the linearized scheme is more time-saving in computation.

A high-order combined finite element/interpolation approach for multidimensional nonlinear generalized Benjamin–Bona–Mahony–Burgers equation

A high-order combined finite element/interpolation approach is developed for solving a multidimensional nonlinear generalized Benjamin–Bona–Mahony–Burgers equation subjected to suitable initial and boundary conditions. In the proposed high-order scheme we approximate the time derivative with piecewise polynomial interpolation of second-order and use the finite element discretization of piecewise polynomials of degree q and q+1, where q≥2 is an integer, to approximating the space derivatives. The stability together with the error estimates of the constructed technique are established in W21(Ω)-norm. The analysis suggests that the developed numerical scheme is unconditionally stable, temporal second-order accurate and convergence in space with order q. Furthermore, the new procedure is faster and more efficient than a broad range of numerical methods discussed in the literature for the given initial–boundary value problem. A wide set of examples are performed to confirming the theoretical studies.

The splitting-based semi-implicit finite-difference schemes for simulation of blood flow in arteries

This paper is devoted to construction and analysis of splitting-based finite-difference schemes for simulation of blood flow in applications of one-dimensional models. Proposed schemes are constructed for inviscid and viscid models of blood. The proposed approach is based on the modification of averaged incompressibility condition, rewritten for the square root of a cross-sectional area. It is approximated by an absolutely stable scheme with a skew-symmetric operator. The constructed schemes can be classified as semi-implicit. The stability of proposed schemes is analyzed by the method of energy inequalities. Sufficient stability conditions are obtained. For the problems with analytical solutions, it is demonstrated that a second-order convergence rate is achieved in practice.The proposed schemes are compared with the widely used explicit finite-difference schemes, such as Lax–Wendroff, Lax–Friedrichs and McCormack schemes. The benchmark problems for simulation of flows in the following vascular systems are considered: human carotid artery, human aorta with bifurcation, vessel with bifurcation and absorbing outflow conditions, and networks with 31, 63, and 127 vessels. It is demonstrated that the computations based on the splitting-based semi-implicit schemes can be performed much faster than for the explicit schemes with close values of relative errors.

Decoupled second-order energy stable scheme for an electrohydrodynamic model with variable electrical conductivity

The motion of ions in the complex fluids widely appears in hydrodynamics, geodynamics and geophysics. Setting up suitable mathematical model, performing accurate and efficient numerical simulations are essential to understand the underlying physical principle in these phenomena. In this paper, we present a linear, second-order accurate in time, decoupled and unconditionally energy stable numerical scheme for solving an electrohydrodynamic power-law model with variable electrical conductivity. Based on a logarithm transformation for conductivity and the zero energy contribution property of the nonlinear coupling terms in the model, we derive an equivalent system by introducing a nonlocal auxiliary variable. The two-step backward differentiation scheme and the finite element method are used for the temporal and spatial discretizations, respectively. To decouple the system, the nonlocal splitting technique is employed to yield several linear subsystems, which can be solved very efficiently. Numerical simulations are carried out to demonstrate the second-order convergence in time and the unconditional energy stability. Moreover, the effects of Coulombic force and power-law fluid exponents are numerically investigated.

A coupled nonlinear finite element scheme for anisotropic diffusion equation with nonlinear capacity term

A new nonlinear finite element method is studied to solve multi-dimensional anisotropic nonlinear diffusion equation with nonlinear capacity term, especially to secure a vivid simulation in the case of problems with transient physical quantities. In the scheme design, fully implicit two-layer coupled discretization is applied to assure high time accuracy, and finite element discretization is applied to assure high space accuracy. By introducing Ritz projection and new inductive reasoning methods, we develop the discrete functional analysis technique from that for finite difference schemes, and establish a new general analysis framework for nonlinear finite element schemes. Consequently, we overcome the difficulties arising from the strong coupling of the nonlinear finite element discretizations for the capacity term and diffusion operator, and prove the existence and boundedness of the nonlinear finite element solution, as well as its second-order time and optimal order space convergence. Numerical experiments and comparisons confirm the theoretical analysis results and demonstrate its high performance.