Space Step Size Research Articles

Ablation and molten layer flow simulation for plate model of SiO2f/SiO2 composite material using particle method

In this paper, the moving particle semi-implicit method (MPS) is extended from calculating free mobility to simulating the extremely viscous and temperature-dependent molten layer flow of SiO2f/SiO2 composite material under aerodynamic heating conditions, which includes strong heating and shear of incoming flow. A method for applying heat flux and airflow shear, based on the conceptual particle approach, has been established. Heat transfer, melting, solidification, and evaporation behaviors are considered, with temperature-dependent viscosity variations also accounted for. The ablative regression of the plate model is verified using experimental results of the SiO2f/SiO2 composite material, and results from convergence analysis demonstrate the accuracy of the space step size selection. Surface morphology analysis through three-dimensional computation indicates that the extended particle method also accurately describes the surface morphology of SiO2f/SiO2 composite material under aerodynamic heating conditions. Thus, the extended particle method accurately simulates both the ablation process and the surface morphology of the SiO2f/SiO2 composite material. The influences of acceleration and surface tension are discussed. Ablative recession, when subject to acceleration, is smaller than that observed in its absence. When exposed to surface tension, the liquid layer tends to form a spherical shape, and the particles behave as a cohesive unit, resulting in smaller ablative recession than in the absence of surface tension.

Non-polynomial spline method for computational study of reaction diffusion system

This work addresses an efficient and new numerical technique utilizing non-polynomial splines to solve system of reaction diffusion equations (RDS). These system of equations arise in pattern formation of some special biological and chemical reactions. Different types of RDS are in the form of spirals, hexagons, stripes, and dissipative solitons. Chemical concentrations can travel as waves in reaction-diffusion systems, where wave like behaviour can be seen. The purpose of this research is to develop a stable, highly accurate and convergent scheme for the solution of aforementioned model. The method proposed in this paper utilizes forward difference for time discretization whereas for spatial discretization cubic non-polynomial spline is used to get approximate solution of the system under consideration. Furthermore, stability of the scheme is discussed via Von-Neumann criteria. Different orders of convergence is achieved for the scheme during a theoretical convergence test. Suggested method is tested for performance on various well known models such as, Brusselator, Schnakenberg, isothermal as well as linear models. Accuracy and efficiency of the scheme is checked in terms of relative error (E R ) and L ∞ norms for different time and space step sizes. The newly obtained results are analyzed and compared with those available in literature.

Variable-step [formula omitted] method combined with time two-grid algorithm for multi-singularity problems arising from two-dimensional nonlinear delay fractional equations

In this paper, we focus on the numerical simulation for two-dimensional nonlinear fractional sub-diffusion equations in the presence of time delay. Firstly, we investigate the existence, uniqueness and regularity of the solution for such problems. The theoretical result implies that the solution at τ+ is smoother than that at 0+, where τ is a constant time delay, and this is an improvement for the work (Tan et al., 2022). Secondly, a high-order difference scheme based on L1 method is constructed. For the sake of repairing the convergence order in temporal direction and improving the computational efficiency, an efficient time two-grid algorithm based on nonuniform meshes is first developed. The convergence order of the two-grid scheme reaches O(NF−min{rα,2−α}+NC−min{2rα,4−2α}+h12+h22), where NF and NC represent the number of the fine and coarse grids respectively, while h1 and h2 are the space-step sizes. Furthermore, stability and convergence analysis of the proposed scheme are carefully verified by energy method. Finally, numerical experiments are carried out to show the validity of theoretical statements.

A second order difference method combined with time two-grid algorithm for two-dimensional time-fractional Fisher equation

In this paper, a finite difference (FD) scheme based on time two-grid algorithm is proposed for solving the two-dimensional time-fractional Fisher equation (2D-TFFE). Firstly, the Caputo fractional derivative and the spatial derivative are discretized by the L 2 − 1 σ formula and the central difference formula, respectively. In order to improve the efficiency of computation, the time two-grid algorithm is then constructed. Secondly, based on the cut-off function technic, stability and convergence of the time two-grid FD scheme are obtained by the energy method, and the global convergence order is O ( τ F 2 + τ C 4 + h x 2 + h y 2 ) , where τ F and τ C represent time-step sizes on the fine and coarse grid, respectively, while h x and h y represent the space-step sizes. Finally, numerical experiments are presented to show the feasibility and efficiency of the algorithm.

Some novel exact and non-standard finite difference schemes for a class of diffusion–advection–reaction equation

In this paper, a generalized diffusion–advection–reaction equation (GDARE) is considered and by using the solitary wave solution two new class of exact finite difference (EFD) schemes are proposed under certain functional relationship between the temporal and space step sizes. Suitable denominator functions are identified from these EFD schemes and a new non-standard finite difference (NSFD) scheme is developed. This NSFD scheme preserves positivity and boundedness properties of the solution of GDARE. The L 2 , L ∞ errors and CPU times are obtained for the proposed NSFD and EFD schemes to show the proficiency and accuracy of the schemes.

Numerical Simulation for Solitary Waves of the Generalized Zakharov Equation Based on the Lattice Boltzmann Method

The generalized Zakharov equation is a widely used and crucial model in plasma physics, which helps to understand wave particle interactions and nonlinear wave propagation in plasma. The solitary wave solution of this equation provides insights into phenomena such as electron and ion acoustic waves, as well as magnetic field disturbances in plasma. The numerical simulation of solitary wave solutions to the generalized Zakharov equation is an interesting problem worth studying. This is crucial for plasma-based technology, as well as for understanding nonlinear wave propagation in plasma physics and other fields. In this study, a numerical investigation of the generalized Zakharov equation using the lattice Boltzmann method has been conducted. The lattice Boltzmann method is a new modeling and simulating method at the mesoscale. A lattice Boltzmann model was constructed by performing Taylor expansion, Chapman–Enskog expansion, and time multiscale expansion on the lattice Boltzmann equation. By defining the moments of the equilibrium distribution function appropriately, the macroscopic equation can be restored. Furthermore, the numerical experiments for the equation are carried out with the parameter lattice size m=100, time step Δt=0.001, and space step size Δx=0.4. The solitary wave solution of the equation is numerically simulated. Numerical results under different parameter values are compared, and the convergence and effectiveness of the model are numerically verified. It is obtained that the model is convergent in time and space, and the convergence orders are all 2.24881. The effectiveness of our model was also verified by comparing the numerical results of different numerical methods. The lattice Boltzmann method demonstrates advantages in both accuracy and CPU time. The results indicate that the lattice Boltzmann method is a good tool for computing the generalized Zakharov equation.

A time two-grid algorithm for two-dimensional nonlinear time-fractional partial integro-differential equations

In this paper, a temporal second order two-grid difference scheme is proposed for the two-dimensional nonlinear time-fractional partial integro-differential equations with a weakly singular kernel. The first-order backward difference and L1 formula are used in the temporal direction to estimate the first level of time, the L2−1σ formula and L1-type formula are used in the temporal direction for later time steps, and the central difference formula is used in the spatial directions. To improve the computational efficiency of nonlinear system, an efficient time two-grid algorithm is proposed. This algorithm firstly solves a nonlinear system on the coarse grid, and then the Lagrangian linear interpolation is applied on the coarse grid to estimate the function values on the fine grid. The stability and convergence of the two-grid difference scheme are analyzed by the energy method. The convergence order of the two-grid difference scheme is O(τF2+τC4+hx2+hy2), where τF and τC are the time step sizes of fine grid and coarse grid respectively, while hx and hy are the space step sizes. Numerical experiments show that the accuracy of the theoretical analysis and the efficiency of the two-grid algorithm.

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

A higher order stable numerical approximation for time‐fractional non‐linear Kuramoto–Sivashinsky equation based on quintic B$$ \mathfrak{B} $$‐spline

This article deals with designing and analyzing a higher order stable numerical analysis for the time‐fractional Kuramoto–Sivashinsky (K‐S) equation, which is a fourth‐order non‐linear equation. The fractional derivative of order present in the considered problem is taken into Caputo sense and approximated using the scheme. In space direction, the discretization process uses quintic ‐spline functions to approximate the derivatives and the solution of the problem. The present approach is unconditionally stable and is convergent with rate of accuracy , where and denote the space and time step sizes, respectively. We have also noted that the linearized version of the K‐S equation leads the rate of accuracy to . The present approach is also highly effective for the time‐fractional Burgers' equation. We have shown that the present approach provides better accuracy than the scheme with the same computational cost for several linear/non‐linear problems, with classical as well as fractional time derivatives.

Richardson extrapolation method for solving the Riesz space fractional diffusion problem

AbstractThe Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods.

Second‐order scalar auxiliary variable Fourier‐spectral method for a liquid thin film coarsening model

In this paper, we propose and analyze a second‐order accurate (in time) numerical scheme for the droplet liquid film coarsening model by using the scalar auxiliary variable (SAV) method, and the Fourier‐spectral method is used in space. The scheme is linear; thus, it could be solved very efficiently. Meanwhile, due to the application of SAV approach, the unconditional energy stability of the numerical scheme is also derived, without any restriction on the time step size. In addition, we also provide a rigorous error estimate that shows that our second‐order numerical scheme with Fourier‐spectral method in space converges with order , where and are time and space step sizes, respectively. Finally, a few numerical experiments are performed, which confirms the efficiency and accuracy of the proposed scheme, including the tests of the convergence, the mass conservation and the decrease of the energy. The simulation of coarsening process with time is also observed.

Conservative EQ1rot nonconforming FEM for nonlinear Schrödinger equation with wave operator

AbstractIn this paper, we consider leap‐frog finite element methods with element for the nonlinear Schrödinger equation with wave operator. We propose that both the continuous and discrete systems can keep mass and energy conservation. In addition, we focus on the unconditional superconvergence analysis of the numerical scheme, the key of which is the time‐space error splitting technique. The spatial error is derived independently with order in ‐norm, where and denote the space and time step size. Then the unconditional optimal error and superclose result with order are deduced, and the unconditional optimal error is obtained with order by using interpolation theory. The final unconditional superconvergence result with order is derived by the interpolation postprocessing technique. Furthermore, we apply the proposed leap‐frog finite element methods to solve the logarithmic Schrödinger equation with wave operator by introducing a regularized system with a small regularization parameter . The detailed theoretical conclusions, including the mass and energy conservation laws of the continuous regularized and discrete regularized systems and the convergence and superconvergence results, are presented as prolongation of the previous work. At last, some numerical experiments are given to confirm our theoretical analysis.

Dynamic Phase‐Field Fracture in Viscoelastic Materials using a First‐Order Formulation

AbstractIn this contribution we present analytical results on a model for dynamic fracture in viscoelastic materials at small strains that have been obtained in full depth in [1]. In the model, the sharp crack interface is regularized with a phase‐field approximation, and for the phase‐field variable a viscous evolution with a quadratic dissipation potential is employed. A non‐smooth penalization prevents material healing. The viscoelastic momentum balance is formulated as a first order system and coupled in a nonlinear way to the non‐smooth evolution equation of the phase field. We give a full discretization in time and space using a discontinuous Galerkin method for the first‐order system. We discuss the existence of discrete solutions and, with the step size in space and time tending to zero, their convergence to a suitable notion of weak solution of the system. Eventually, we provide a numerical benchmark and compare it with simulation results found in [2].

High-order numerical algorithm and error analysis for the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation

In this paper, we first construct an appropriate new generating function, and then based on this function, we establish a fourth-order numerical differential formula approximating the Riesz derivative with order γ∈(1,2]. Subsequently, we apply the formula to numerically study the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation and obtain a difference scheme with convergence order Oτ2+hx4+hy4, where τ denotes the time step size, hx and hy denote the space step sizes, respectively. Furthermore, with the help of some newly derived discrete fractional Sobolev embedding inequalities, the unique solvability, the unconditional stability, and the convergence of the constructed numerical algorithm under different norms are proved by using the discrete energy method. Finally, some numerical results are presented to confirm the correctness of the theoretical results and verify the effectiveness of the proposed scheme.

An operator splitting method for multi-asset options with the Feynman-Kac formula

In this paper, we propose an unconditionally stable numerical technique for a multi-dimensional Black-Scholes equation to price an option with high accuracy. The proposed scheme uses the operator splitting method to reduce the multi-dimensional partial differential equation to a set of one-dimensional sub-problems. The computational domain is discretized with a uniform space and time step size to approximate the option values and asset correlation-related terms by piecewise quadratic polynomials. In order to obtain the numerical solutions of the sub-problems, we analytically integrate the polynomials to estimate the expectation of the Feynman-Kac formula. We compare our method with the implicit operator splitting method (OSM), a widely used finite difference method in the industry. Numerical experiments show that the proposed method outperforms OSM in terms of convergence in space and time directions. We also provide analysis to guarantee the unconditional stability of our method by exploiting the Feynman-Kac recursively.

Temporal high-order, unconditionally maximum-principle-preserving integrating factor multi-step methods for Allen-Cahn-type parabolic equations

We develop a class of explicit unconditionally maximum-principle-preserving multi-step methods for the Allen-Cahn-type semilinear parabolic equations. Based on the usual second-order finite difference space discretization, we first show that the strong-stability-preserving (SSP) integrating factor multi-step method can preserve the maximum-principle under the condition τ≤CτFE, where C is the SSP coefficient and τFE is independent of the space step size, thus avoiding the strict parabolic Courant-Friedrichs-Lewy condition τ=O(h2). By introducing the stabilization technique and replacing the exponential functions with two different approximations, we obtain two types of improved stabilized integrating factor multi-step (isIFMS) schemes. The unconditional maximum-principle-preservation, mass-conservation, linear stability, and error estimates are analyzed for the isIFMS schemes. Since explicit SSP multi-step methods have no order barrier, the proposed framework offers a concise but effective approach for developing temporal high-order, unconditionally maximum-principle-preserving schemes for Allen-Cahn-type equations, and mass-conserving schemes for their conservative reformulations. Finally, numerical experiments validate theoretical results of the proposed isIFMS schemes.

Local discontinuous Galerkin method for multi-term variable-order time fractional diffusion equation

This paper presents an effective numerical method for multi-term variable-order time fractional diffusion equations with the variable-order fractional derivative. The local discontinuous Galerkin method and the finite difference method are used in the spatial and temporal directions, respectively. We prove that the scheme is unconditional stable and convergent with O(hs+1+(Δt)2−r), where r=max{ɛ(t)}. s, h, Δt are the degree of piecewise polynomials, the space step sizes, and the time step sizes, respectively. Some numerical experiments are used to illustrate the effectiveness and applicability of the scheme.

Analysis of Two-Level Mesh Method for Partial Integro-Differential Equation

In this paper, we present two-level mesh scheme to solve partial integro-differential equation. The proposed method is based on a finite difference method. For the first step, we use finite difference method in time and global radial basis function (RBF) finite difference (FD) in space. For the second step, we use the finite difference method to solve the proposed problem. This two-level mesh scheme is obtained by combining the radial basis function with finite difference. We prove the stability and convergence of scheme and show that the convergence order is O τ 2 + h 2 , where τ and h are the time step size and space step size, respectively. The results of numerical examples are compared with analytical solutions to show the efficiency of proposed scheme. The numerical results are in good agreement with theoretical ones.

Numerical approximation of time fractional partial integro-differential equation based on compact finite difference scheme

In this paper, a new numerical scheme based on weighted and shifted Grünwald formula and compact difference operate is proposed. The proposed numerical scheme is used to solve time fractional partial integro-differential equation with a weakly singular kernel. Meanwhile the time fractional derivative is denoted by the Riemann-Liouville sense. Subsequently, we prove the stability and convergence of the mentioned numerical scheme and show that the numerical solution converges to the analytical solution with order O(τ2 + h4), where τ and h are time step size and space step size, respectively. The advantage is that the accuracy of the suggested schemes is not dependent on the fractional α. Furthermore, the numerical example shows that the method proposed in this paper is effective, and the calculation results are consistent with the theoretical analysis.

A Time Two-Mesh Compact Difference Method for the One-Dimensional Nonlinear Schrödinger Equation.

The nonlinear Schrödinger equation is an important model equation in the study of quantum states of physical systems. To improve the computing efficiency, a fast algorithm based on the time two-mesh high-order compact difference scheme for solving the nonlinear Schrödinger equation is studied. The fourth-order compact difference scheme is used to approximate the spatial derivatives and the time two-mesh method is designed for efficiently solving the resulting nonlinear system. Comparing to the existing time two-mesh algorithm, the novelty of the new algorithm is that the fine mesh solution, which becomes available, is also used as the initial guess of the linear system, which can improve the calculation accuracy of fine mesh solutions. Compared to the two-grid finite element methods (or finite difference methods) for nonlinear Schrödinger equations, the numerical calculation of this method is relatively simple, and its two-mesh algorithm is implemented in the temporal direction. Taking advantage of the discrete energy, the result with in the discrete -norm is obtained. Here, and are the temporal parameters on the coarse and fine mesh, respectively, and h is the space step size. Finally, some numerical experiments are conducted to demonstrate its efficiency and accuracy. The numerical results show that the new algorithm gives highly accurate results and preserves conservation laws of charge and energy. Furthermore, by comparing with the standard nonlinear implicit compact difference scheme, it can reduce the CPU time without loss of accuracy.