Abstract: This article presents a meshless method to solve three-dimensional elliptic interface problem. The method is based on the generalized finite difference method, which expresses the derivatives of unknown variables by linear combinations of nearby function values. The proposed method turns the interface problem into some boundary value sub-problems coupled by the interface conditions. This conversion leads to an important feature, that is, the proposed method is not sensitive to the jump coefficients or the interface's geometry. It can handle different complex interfaces by only changing the level set function of the interface in the process. Several numerical examples with sufficient complexity verify the accuracy and stability of the method. For some given examples, the method is more accurate than the classical immersed finite element method.

Abstract: A summation-by-parts simultaneous approximation term (SBP-SAT) finite-difference time-domain (FDTD) subgridding method is proposed to model geometrically fine structures in this article. Compared with other works, the proposed SBP-SAT FDTD method uses the staggered Yee’s grid without adding or modifying any field components through field extrapolation on the boundaries to make the discrete operators satisfy the SBP property. The accuracy of extrapolation keeps consistency with that of the second-order finite-difference scheme near the boundaries. In addition, the SATs are used to weakly enforce the tangential boundary conditions between multiple mesh blocks with different mesh sizes. With carefully designed interpolation matrices and selected free parameters of the SATs, no dissipation occurs in the whole computational domain. Therefore, its long-time stability is theoretically guaranteed. Four numerical examples are carried out to validate its effectiveness. Results show that the proposed SBP-SAT FDTD subgridding method is stable, accurate, efficient, and easy to implement based on the existing FDTD codes with only a few modifications.

Abstract: A numerical study of a two-parameter singularly perturbed time-delay parabolic equation has been initiated. The proposed technique is based on a fitted operator finite difference scheme. The first step involves the discretization of the time variables using the Crank–Nicolson method. This results in singularly perturbed semi-discrete problems, which are further discretized in space using the exponentially fitted tension-spline finite difference method. The error analysis is carried out. It is shown to be parameter-uniformly convergent and second-order accurate. To support the theoretical results, numerical experiments are carried out by applying the proposed technique to two test examples.

Abstract: Heat transfer phenomenon occurs in a wide range of industrial and engineering problems. In this framework, investigating the heat transfer behavior using an efficient mathematical tool is the hot research area in the research community. Therefore, we discussed the impact of fractional Maxwell hybrid nanofluid on heat transfer under the effects of Joules Heating, Magnetic field, and viscous dissipation via porous Medium with Caputo time fractional derivative. We have considered an infinite vertical wall with adaptable temperature moving with jerked motion along x-direction. Water is taken as base fluid and nano-sized particles of Copper (Cu) & Alumina (Al2O3) are used for the preparation of nanofluid. The governing equations of nonlinear Caputo time fractional model are converted into dimensionless form. A finite difference scheme is developed and applied successfully to get numerical solutions to the problem and to examine the influence of Maxwell hybrid nanofluid, physical parameters, and fractional parameters on velocity and temperature profiles. It can also be observed that the extended finite difference algorithm is very efficient and gives the accurate solution of the discussed model. It can be extended for more numerous types of heat transfer problems arising in physical nature with complex geometry.

Abstract: In this article, a 3-D hybrid Maxwell’s equations finite-difference time-domain (ME-FDTD)/wave equation-based finite-element time-domain (WE-FETD) method is proposed. This method retains the nonconformal mesh and the implicit–explicit time integration scheme. The WE-FETD region is based on the wave equation rather than Maxwell’s curl equations. This method needs to store all the electric fields in the entire region and only the magnetic fields on the interface, which can prominently reduce degrees of freedom (DoFs) and save calculation time. The ME-FDTD region follows Yee’s scheme. A Maxwell’s equations spectral element time-domain (ME-SETD) region and a virtual region are used to combine the ME-FDTD and WE-FETD regions. Consequently, a WE-FETD/ME-SETD/Virtual/ME-FDTD framework is formed. Hybrid Newmark-beta (NB) and Crank–Nicolson (CN) time stepping are employed for implicit WE-FETD and ME-SETD regions. The leapfrog (LF) time integration is used for the explicit virtual and FDTD regions. At the interface, it employs upwind flux in the discontinuous Galerkin (DG) method to couple neighboring regions. Numerical examples are included to demonstrate the accuracy of the proposed method. Several cases exhibit the improved efficiency compared with the hybrid FDTD/FETD method only based on Maxwell’s equations and the pure FETD method.

Abstract: The marine environment can be considered a fluid-solid coupled media governed by the acoustic-elastic coupled equation (AECE). For complex fluid-solid coupled models with irregular seabeds, the well-known finite difference method (FDM) often causes ripple-shaped scattering due to the rectangular grid. To overcome above problem, a meshless method (MM), called generalized finite-difference method (GFDM), for irregular node distributions is developed and applied to AECE forward modeling by employing a body-fitted node-generation method (BNGM), which provides robust node distributions for GFDM. Compared with the finite-difference method (FDM), the proposed GFDM improves modeling accuracy and eliminates divergence. Furthermore, the proposed MM was applied to an AECE-based elastic full waveform inversion (EFWI). Owing to the well-matched irregular seabed, the proposed EFWI can invert the accurate elastic model for both ocean bottom cable (OBC) and pressure data. Numerical examples demonstrate that this method can efficiently improve the numerical accuracy and build an accurate velocity model even using noisy data, which is helpful for high-precision seismic exploration in the complex marine environments.

Abstract: This paper applied the space-time generalized finite difference scheme for solving the nonlinear dispersive shallow water waves as the modified Camassa−Holm equation, modified Degasperis-Procesi equation, Fornberg-Whitham equation, and its modified form. The proposed meshless numerical scheme was composed of the space-time generalized finite difference method, the two-step Newton-Raphson method, and the time-marching method. The space-time approach can treat the temporal derivative as one of the spatial derivatives. This numerical technique enables all the partial derivatives in the governing equation can be discretized by a spatial discretization method, and the mixed derivative can efficiently deal with using the proposed meshless numerical scheme. The space-time generalized finite difference method is advanced from the Taylor series expansion and the moving-least square method. The numerical discretization process is only related to functional data and weighting coefficients on the central node and its nearby nodes. Thus, the matrix system composed of nonlinear algebraic equations will be a sparse matrix and can be efficiently solved by the two-step Newton-Raphson method. Furthermore, the time-marching method was utilized to proceed with the space-time domain along the time axis. In this paper, several numerical examples were tested to verify the capability of the proposed space-time generalized finite difference scheme.

Abstract: This letter presents a higher order finite-difference (FD) and Padé approximations method for the three-dimensional (3-D) parabolic equation (PE) to predict radio-wave propagation. This method uses a fourth-order FD approximation of the differential operator in the transverse direction and a higher order Padé approximation of the operator in the propagation direction. The fourth-order FD and higher order Padé (4FDHP) method is then derived. The Leontovich impedance boundary for the 4FDHP method and boundary for the second-order FD and higher order Padé (2FDHP) method are also derived. The important problem of the propagation angle of the different approximations for the 3D-PE is investigated. Simulated results show that the proposed 4FDHP method achieves a larger propagation angle and higher accuracy than those of the 2FDHP method and the Mitchell-Fairweather alternative-direction-implicit method.

Abstract: This paper presents a new numerical computation code based on the Collocation Spectral Method (CSM) to solve the partial differential equation describing the potential distribution along a polluted insulator. The pollution on insulator surface is modeled by a uniform thin layer. The results of the electric potential distribution obtained by CSM are compared with those of the Finite Difference Method, the Finite Element Method and the analytical solution. The accuracy of this method in terms of maximum (error) is discussed. Based on the newly developed code, the influence of physical parameters of the insulator model, such as the relative permittivity, surface conductivity and insulator thickness on the electric potential distribution are investigated.

Abstract: This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, a family of non-standard methods is derived to deal with a system of ordinary differential equations (ODEs) arising from the semi-discretization of initial-boundary value partial differential equations (PDEs). Further, a method of this family is considered as a special case and coupled with a fourth-order compact finite difference resulting in a combined numerical scheme to solve initial-boundary value PDEs. The combined scheme has first-order accuracy in time and fourth-order accuracy in space. Some basic characteristics of the scheme are analysed and a section concerning the numerical experiments is presented demonstrating the good performance of the combined numerical scheme.

Abstract: This paper addresses the solution of a differential-difference type equation having an interior layer behaviour. A difference scheme is suggested to solve this equation using a non-standard finite difference method. Finite differences are derived from the first and second order derivatives. Using these approximations, the given equation is discretized. The discretized equation is solved using the algorithm for the tridiagonal system. The method is examined for convergence. Numerical examples are illustrated to validate the method. Maximum errors in the solution, in contrast to the other methods are organized to justify the method. The layer behaviour in the solution of the examples is depicted in graphs.

Abstract: Even when the wave equation is a classical problem to solve in courses about hyperbolic partial differential equations, the numerical solution for this equation in irregular regions has been little studied. This is because of all the problems that could arise when solving the second-order derivative in time. This paper presents a generalized finite difference scheme applied to the numerical solution of the wave equation, solved on clouds of points for highly irregular domains. The implementation over clouds of points allows this scheme to compute satisfactory results, with minor quadratic errors, over highly irregular regions, as discussed in the present manuscript. Also, the numerical results show the accuracy obtained with a straightforward and low-cost implementation.

Abstract: A new explicit and nondissipative finite-difference time-domain (FDTD) method in two and three dimensions is proposed for the relaxation of the Courant condition. The third-degree spatial difference terms with second- and fourth-order accuracies are added with coefficients to the time-development equations of FDTD(2,4). Optimal coefficients are obtained by a brute-force search of the dispersion relations, which reduces phase velocity errors but satisfies the numerical stabilities as well. The new method is stable with large Courant numbers, whereas the conventional FDTD methods are unstable. The new method also has smaller numerical errors in the phase velocity than conventional FDTD methods with small Courant numbers.

Abstract: In this study, a high-order compact finite difference method is used to solve Lane–Emden equations with various boundary conditions. The norm is to use a first-order finite difference scheme to approximate Neumann and Robin boundary conditions, but that compromises the accuracy of the entire scheme. As a result, new higher-order finite difference schemes for approximating Robin boundary conditions are developed in this work. We test the applicability and performance of the method using different examples of Lane–Emden equations. Convergence analysis is provided, and it is consistent with the numerical results. The results are compared with the exact solutions and published results from other methods. The method produces highly accurate results, which are displayed in tables and graphs.

Abstract: PurposeWhen modeling heat propagation in biological bodies, a non-negligible relaxation time (typically between 15-30 s) is required for the thermal waves to accumulate and transfer, i.e. thermal waves propagate at a finite velocity. To accommodate for this feature that is characteristic to heat transfer in biological bodies, the classical Fourier's law has to be modified resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the space-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from final time temperature measurements.Design/methodology/approachThe non-linear and ill-posed blood perfusion coefficient identification problem is reformulated as a non-linear minimization problem of a Tikhonov regularization functional subject to lower and upper simple bounds on the unknown coefficient. For the numerical discretization, an unconditionally stable direct solver based on the Crank–Nicolson finite difference scheme is developed. The Tikhonov regularization functional is minimized iteratively by the built-in routine lsqnonlin from the MATLAB optimization toolbox. Both exact and numerically simulated noisy input data are inverted.FindingsThe reconstruction of the unknown blood perfusion coefficient for three benchmark numerical examples is illustrated and discussed to verify the proposed numerical procedure. Moreover, the proposed algorithm is tested on a physical example which consists of identifying the blood perfusion rate of a biological tissue subjected to an external source of laser irradiation. The numerical results demonstrate that accurate and stable solutions are obtained.Originality/valueAlthough previous studies estimated the important thermo-physical blood perfusion coefficient, they neglected the wave-like nature of heat conduction present in biological tissues that are captured by the more accurate thermal-wave model of bio-heat transfer. The originalities of the present paper are to account for such a more accurate thermal-wave bio-heat model and to investigate the possibility of determining its space-dependent blood perfusion coefficient from temperature measurements at the final time.

Abstract: The determination of the natural frequencies for graphene scatterers is attained in the present work using a finite-difference scheme. The frequency-dispersive, two-dimensional material is treated as an equivalent surface current density, while an appropriately derived auxiliary differential equation is introduced to acquire a linear eigenvalue problem. Then, the finite-difference discretization is performed in a Yee-cell manner to straightforwardly implement all the required curl operations. The proposed scheme is validated using a rectangular graphene patch at the THz regime, where the surface plasmon polariton waves of the two-dimensional material are supported. The numerically extracted eigenstates are compared with the absorption cross-section analysis of full-wave simulations, with respect to the resonance frequencies. The obtained results substantiate the precise modeling, while, also, retrieving the quality factor of the supported modes.

Abstract: We present a finite-difference based immersed interface method for the high-order discretization of 3D advection-diffusion problems on regular Cartesian grids. Our approach efficiently handles convex and non-convex geometries using a weighted least squares polynomial reconstruction algorithm. We analyze the stability of the approach for 2D and 3D parabolic and hyperbolic problems and demonstrate stable convergence results at third-order for advection and at fourth and sixth order for diffusion problems. Our immersed interface approach naturally handles one-sided Dirichlet or Neumann boundary conditions as well as two-sided jump boundary conditions within the same framework, opening the door to high-order treatment of 3D interface-coupled multiphysics problems. We demonstrate the capability of our approach using a 3D conjugate heat transfer problem resolved with third-order accuracy on a multi-resolution adaptive grid.

Abstract: In this paper, the numerical method for a multiterm time-fractional reaction–diffusion equation with classical Robin boundary conditions is considered. The full discrete scheme is constructed with the L1-finite difference method, which entails using the L1 scheme on graded meshes for the temporal discretisation of each Caputo fractional derivative and using the finite difference method on uniform meshes for spatial discretisation. By dealing with the discretisation of Robin boundary conditions carefully, sharp error analysis at each time level is proven. Additionally, numerical results that can confirm the sharpness of the error estimates are presented.

Abstract: The symmetric regularized long wave (SRLW) equation is a mathematical model used in many areas of physics; the solution of the SRLW equation can accurately describe the behavior of long waves in shallow water. To approximate the solutions of the equation, a time two-mesh (TT-M) decoupled finite difference numerical scheme is proposed in this paper to improve the efficiency of solving the SRLW equation. Based on the time two-mesh technique and two time-level finite difference method, the proposed scheme can calculate the velocity u(x,t) and density ρ(x,t) in the SRLW equation simultaneously. The linearization process involves a modification similar to the Gauss-Seidel method used for linear systems to improve the accuracy of the calculation to obtain solutions. By using the discrete energy and mathematical induction methods, the convergence results with O(τC2+τF+h2) in the discrete L∞-norm for u(x,t) and in the discrete L2-norm for ρ(x,t) are proved, respectively. The stability of the scheme was also analyzed. Finally, some numerical examples, including error estimate, computational time and preservation of conservation laws, are given to verify the efficiency of the scheme. The numerical results show that the new method preserves conservation laws of four quantities successfully. Furthermore, by comparing with the original two-level nonlinear finite difference scheme, the proposed scheme can save the CPU time.