Implicit Difference Scheme Research Articles

High-order blended compact difference schemes for the 3D elliptic partial differential equation with mixed derivatives and variable coefficients

Blended compact difference (BCD) schemes with fourth- and sixth-order accuracy are proposed for approximating the three-dimensional (3D) variable coefficients elliptic partial differential equation (PDE) with mixed derivatives. With truncation error analyses, the proposed BCD schemes can reach their theoretical accuracy, respectively, for the interior gird points and require 19 points compact stencil. They fully blend the implicit compact difference (CD) scheme and the explicit CD scheme together to make the derivation method and programming easier. The BCD schemes are also decoupled, which means the unknown function and its derivatives are separately resolved by different finite difference equations. Moreover, the sixth-order schemes are developed to solve the first-order derivatives, the second-order derivatives and the second-order mixed derivatives on boundaries. Several test problems are applied to show that the present BCD schemes are more accurate than those in the literature.

Cylindrical body temperature field simulation which made in the transitional thermal process conditions out of polymer material

The advancing requirements for strength, relaxation, thermophysical, electrical, and other structural elements characteristics actualizes the polymer composite material use for the soft part and node point manufacture, which improves performance index. This paper reported the need to take into account relaxation phenomena in predicting the body’s thermal field development that is made of polymeric materials, and the thermal relaxation time and the thermal damping time proportional to the duration of transient thermal process certain periods. In this article three-period thermal process in a cylindrical body mathematical model is presented. cylindrical body made of a low-heat-conducting material by using a heat conduction hyperbolic equation that is reflecting the heat flow relaxation and thermal damping phenomenon. A numerical solution to the problem of unsteady heat conduction in a circular disk for a two-phase delay equation is presented, which is based on the grid method implementation by using a three-layer implicit difference scheme and the finite difference method use. Calculation formulas for the run-through coefficients as well as the temperature values at the outer boundaries are concluded using the boundary conditions approximation for the intermediate and upper time layers, taking into account the multi-period of the process. The implementation of the modified run-through method when solving the non-stationary heat conduction problem in a cylindrical body, taking into account the finite heat propagation speed and thermal damping is described. The calculation results for the cylindrical body temperature field are obtained by using the polymethyl methacrylate example upon sudden heating based on a model with a two-phase delay. The results presented in this paper aid in an increase in predicting temperature field accuracy in polymer composite materials in the transient thermal processes study.

An Efficient Numerical Scheme for Variable-Order Fractional Sub-Diffusion Equation

The variable-order (VO) fractional calculus can be seen as a natural extension of the constant-order, which can be utilized in physical and biological applications. In this study, we derive a new numerical approximation for the VO fractional Riemann–Liouville integral formula and developed an implicit difference scheme (IDS) for the variable-order fractional sub-diffusion equation (VO-FSDE). The derived approximation used in the VO time fractional derivative with the central difference approximation for the space derivative. Investigated the unconditional stability by the van Neumann method, consistency, and convergence analysis of the proposed scheme. Finally, a numerical example is presented to verify the theoretical analysis and effectiveness of the proposed scheme.

ON Numerical Blow-Up Solutions of Semilinear Heat Equations

This paper is concerned with the numerical blow-up solutions of semi-linear heat equations, where the nonlinear terms are of power type functions, with zero Dirichlet boundary conditions. We use explicit linear and implicit Euler finite difference schemes with a special time-steps formula to compute the blow-up solutions, and to estimate the blow-up times for three numerical experiments. Moreover, we calculate the error bounds and the numerical order of convergence arise from using these methods. Finally, we carry out the numerical simulations to the discrete graphs obtained from using these methods to support the numerical results and to confirm some known blow-up properties for the studied problems.

An implicit difference scheme for the time-fractional Cahn–Hilliard equations

In this paper, an efficient finite difference scheme is developed for solving the time-fractional Cahn–Hilliard equations which is the well-known representative of phase-field models. The time Caputo derivative is approximated by the popular L1 formula. The stability and convergence of the finite difference scheme in the discrete L2-norm are proved by the discrete energy method. To compare and observe the phenomenon of solution, a generalized difference scheme based on the graded mesh in time is also given. The dynamics of the solution and accuracy of the schemes are verified numerically. Numerical experiments show that the solution of the time-fractional Cahn-Hilliard equation always tends to be in an equilibrium state with the increase of time for different values of order α∈(0,1), which is consistent with the phase separation phenomenon.

Discrete almost maximal regularity and stability for fractional differential equations in Lp([0, 1], Ω)

The present paper is devoted to the study of discrete almost maximal regularity and stability of the difference schemes of nonhomogeneous fractional evolution equations. Using the discretization method of the fractional derivative proposed by Ashyralyev, which actually is the same as the Grünwald-Letnikov approximation for the fractional derivative, the discrete almost maximal regularity and stability of the implicit difference scheme in Lτnp([0,1],Ωn) spaces are established. For the explicit difference scheme, the expression of the solution is obtained. Then the discrete almost maximal regularity and stability of the explicit difference scheme in Lτnp([0,1],Ωn) spaces are achieved as well.

Fast implicit difference schemes for time‐space fractional diffusion equations with the integral fractional Laplacian

In this paper, we develop two fast implicit difference schemes for solving a class of variable‐coefficient time–space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded L1 formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via the M‐matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use the fast sum‐of‐exponential approximation and Toeplitz matrix algorithms to reduce the computational cost for the nonlocal property of time and space fractional derivatives, respectively. The fast schemes greatly reduce the computational work of solving the discretized linear systems from by a direct solver to per preconditioned Krylov subspace iteration and a memory requirement from 𝒪(MN2) to 𝒪(NNexp), where N and (Nexp ≪) M are the number of spatial and temporal grid nodes, respectively. The spectrum of preconditioned matrix is also given for ensuring the acceleration benefit of circulant preconditioners. Finally, numerical results are presented to show the utility of the proposed methods.

A parallel hybrid implementation of the 2D acoustic wave equation

Abstract In this paper, we propose a hybrid parallel programming approach for a numerical solution of a two-dimensional acoustic wave equation using an implicit difference scheme for a single computer. The calculations are carried out in an implicit finite difference scheme. First, we transform the differential equation into an implicit finite-difference equation and then using the alternating direction implicit (ADI) method, we split the equation into two sub-equations. Using the cyclic reduction algorithm, we calculate an approximate solution. Finally, we change this algorithm to parallelize on graphics processing unit (GPU), GPU + Open Multi-Processing (OpenMP), and Hybrid (GPU + OpenMP + message passing interface (MPI)) computing platforms. The special focus is on improving the performance of the parallel algorithms to calculate the acceleration based on the execution time. We show that the code that runs on the hybrid approach gives the expected results by comparing our results to those obtained by running the same simulation on a classical processor core, Compute Unified Device Architecture (CUDA), and CUDA + OpenMP implementations.

Discrete transparent boundary conditions for the equation of rod transverse vibrations

Local perturbations of an infinitely long rod travel to infinity. On the contrary, in the case of a finite length of the rod, the perturbations reach its boundary and are reflected. The boundary conditions constructed here for the implicit difference scheme imitate the Cauchy problem and provide almost no reflection. These boundary conditions are non-local with respect to time, and their practical implementation requires additional calculations at every time step. To minimise them, a special rational approximation, similar to the Hermite - Padé approximation is used. Numerical experiments confirm the high “transparency” of these boundary conditions and determine the conditional stability regions for finite-difference scheme.

Stability of Implicit Difference Schemes for a Linearized Hyperbolic Quasi-Gasdynamic System of Equations

We consider a multidimensional hyperbolic quasi-gasdynamic system of differential equations of the second order in time and space linearized on a constant solution (with an arbitrary velocity). For the linearized system with constant coefficients, we study the implicit three-level weighted and two-level vector difference schemes. The important domination property of the operator of viscous terms (with no allowance for the relaxation parameter) over the operator of convective terms is derived. We apply this property to prove by the energy method that, regardless of the Mach number, our implicit schemes on a nonuniform rectangular mesh (without any conditions on the mesh steps) are stable with respect to the initial data and the right-hand side uniformly in time and the relaxation parameter.

Interdiffusion in many dimensions: mathematical models, numerical simulations and experiment

Over the last two decades, there have been tremendous advances in the computation of diffusion and today many key properties of materials can be accurately predicted by modelling and simulations. In this paper, we present, for the first time, comprehensive studies of interdiffusion in three dimensions, a model, simulations and experiment. The model follows from the local mass conservation with Vegard’s rule and is combined with Darken’s bi-velocity method. The approach is expressed using the nonlinear parabolic–elliptic system of strongly coupled differential equations with initial and nonlinear coupled boundary conditions. Implicit finite difference methods, preserving Vegard’s rule, are generated by some linearization and splitting ideas, in one- and two-dimensional cases. The theorems on the existence and uniqueness of solutions of the implicit difference schemes and the consistency of the difference methods are studied. The numerical results are compared with experimental data for a ternary Fe-Co-Ni system. A good agreement of both sets is revealed, which confirms the strength of the method.

The pointwise estimates of a conservative difference scheme for Burgers' equation

AbstractIn this article, we are concerned with the numerical analysis of a nonlinear implicit difference scheme for Burgers' equation. A priori estimation of the analytical solution is provided in the sense of L∞‐norm when the initial value is bounded in H1‐norm. Conservation, boundedness, and unique solvability are proved at length. Inspired by the method of the priori estimation for the analytical solution, we prove the convergence and stability of the difference scheme in L∞‐norm. Finally, numerical examples are carried out to verify our theoretical results.

Fourth-Order Difference Approximation for Time-Fractional Modified Sub-Diffusion Equation

Fractional differential equations describe nature adequately because of the symmetry properties which describe physical and biological processes. In this article, a fourth-order new implicit difference scheme is formulated and applied to solve the two-dimensional time-fractional modified sub-diffusion equation involving two times Riemann–Liouville fractional derivatives. The stability of the fourth-order implicit difference scheme is investigated using the von Neumann technique. The proposed scheme is shown to be unconditionally stable. Numerical examples are given to illustrate the feasibility of the proposed scheme.

Approximation of Quasilinear Hyperbolic Problems with Discontinuous Coefficients: An Optimal Error Estimate

We develop a theoretical framework for the approximation of a class of second-order quasilinear hyperbolic interface problems on quadratic finite element. Time discretization based on linearized implicit difference scheme with degree two polynomials for interface approximation is proposed. Sufficient conditions, on the input data, that guarantee the existence of a unique solution are given. Under these assumptions, the stability of the scheme is established and convergence rate of optimal order is proved. It is assumed that the interface is arbitrary but smooth.

Highly accurate compact difference scheme for fourth order parabolic equation with Dirichlet and Neumann boundary conditions: Application to good Boussinesq equation

In this work, a three-level implicit compact difference scheme for the generalised form of fourth order parabolic partial differential equation is developed. The discretization is derived by approximating the lower order derivative terms using the governing differential equation with the imbedding technique and is fourth order accurate in space and second order accurate in time. The current approach is advantageous since the boundary conditions are completely satisfied and no further approximations are required to be carried out at the boundaries. The ability of the proposed scheme in handling linear singular problems is examined. The value of first order space derivative is computed alongwith the solution so it does not have to be estimated using the calculated value of the solution. The method successfully works for the highly nonlinear good Boussinesq equation for which more accurate solutions are obtained for the single and the double-soliton solutions in comparison with the existing numerical methods.

Coupled 3D neutron kinetics/thermal-hydraulics calculation of PWR core using nodal Green's function method on Neumann boundary condition

In this paper, we proposed a numerical method to perform the coupled three dimensional (3D) neutron kinetics/thermal-hydraulics calculations for the pressurized water reactor (PWR) core and verified its accuracy. In the spatial neutron kinetics, the nodal Green's function method (NGFM) on Neumann boundary condition is applied for the space discretization procedure, and the implicit difference scheme is used for the time differential. It is assumed that the fission source term in the delayed neutron precursor concentration equation is linearly changed within each time step. For the thermal-hydraulic analysis, the single channel model in which a fuel assembly corresponds to one average thermal-hydraulic channel is applied, and cross flow between channels is not taken into account. And we considered the coolant as a single phase incompressible fluid. The model proposed in here was tested by comparison with the two dimensional (2D) TWIGL problem and the NEACRP-L-335 transient benchmark for the 3D PWR core. The numerical results are in good agreement with reference solutions.

An implicit difference scheme with the KPS preconditioner for two-dimensional time–space fractional convection–diffusion equations

This paper deals with the numerical computation and analysis for a class of two-dimensional time–space fractional convection–diffusion equations. An implicit difference scheme is derived for solving this class of equations. It is proved under some suitable conditions that the derived difference scheme is stable and convergent. Moreover, the convergence orders of the scheme in time and space are also given. In order to accelerate the convergence rate, by combining the Kronecker product splitting (KPS) preconditioner with the generalized minimal residual (GMRES) method, a preconditioning strategy for implementing the difference scheme is introduced. Finally, several numerical examples are presented to illustrate the computational accuracy and efficiency of the methods.

A second-order implicit difference scheme for the nonlinear time-space fractional Schrödinger equation

In this paper, we develop an implicit difference method for solving the nonlinear time-space fractional Schrödinger equation. The scheme is constructed by using the L2-1σ formula to approximate the Caputo fractional derivative, while the weighted and shifted Grünwald formula is adopted for the spatial discretization. The stability and unique solvability of the difference scheme are analyzed in detail. Moreover, we prove that the numerical solution is convergent with second-order accuracy in both temporal and spatial directions. Finally, a linearized iterative algorithm is provided and some numerical tests are presented to validate our theoretical results.

Grünwald Implicit Solution of One-Dimensional Time-Fractional Parabolic Equations Using HSKSOR Iteration

This paper presents the application of a half-sweep iteration concept to the Grünwald implicit difference schemes with the Kaudd Successive Over-Relaxation (KSOR) iterative method in solving one-dimensional linear time-fractional parabolic equations. The formulation and implementation of the proposed methods are discussed. In order to validate the performance of HSKSOR, comparisons are made with another two iterative methods, full-sweep KSOR (FSKSOR) and Gauss-Seidel (FSGS) iterative methods. Based on the numerical results of three tested examples, it shows that the HSKSOR is superior compared to FSKSOR and FSGS iterative methods.

Two-layer finite-difference schemes for the Korteweg-de Vries equation in Euler variables

A family of weighted two-layer finite-difference schemes is presented. Using the example of the numerical solution of model problems on the propagation of a single soliton and the interaction of two solitons, the high quality of explicit-implicit schemes of the Crank-Nichols type with the parameter σ = 0.5 and the order of approximation O(Δt2 + Δx2) isshown. Completely implicit two-layer difference schemes with the parameter σ = 1 and O (Δt+ Δx2) are characterized by absolute stability with a low solution accuracy due to a highapproximation error. The family of explicitly implicit difference schemes is absolutely unstable if the explicitness parameter σ <0.5 prevails. Analysis of the structure of the approximation error, performed using the modified equation method, confirmed the results of numerical simulation.