Accurate Finite Difference Method Research Articles

A residual-based artificial viscosity finite difference method for scalar conservation laws

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.

Elastic Wave Propagation in Curvilinear Coordinates with Mesh Refinement Interfaces by a Fourth Order Finite Difference Method

We develop a fourth order accurate finite difference method for the three dimensional elastic wave equation in isotropic media with the piecewise smooth material property. In our model, the materia...

Wavelet-optimized compact finite difference method for convection–diffusion equations

Abstract In this article, compact finite difference approximations for first and second derivatives on the non-uniform grid are discussed. The construction of diffusion wavelets using compact finite difference approximation is presented. Adaptive grids are obtained for non-smooth functions in one and two dimensions using diffusion wavelets. High-order accurate wavelet-optimized compact finite difference (WOCFD) method is developed to solve convection–diffusion equations in one and two dimensions on an adaptive grid. As an application in option pricing, the solution of Black–Scholes partial differential equation (PDE) for pricing barrier options is obtained using the proposed WOCFD method. Numerical illustrations are presented to explain the nature of adaptive grids for each case.

Simulating the near-field pulse-like ground motions of the Imperial Valley, California, earthquake

Long-period pulse-like ground motions in the near-field are likely to induce severe damage to large structures due to resonance at the natural frequencies of engineered construction. To investigate the origin and characteristics of large velocity pulses from the seismic source, this study identifies 14 stations with velocity pulses from 32 strong-motion seismographs located in the near-field of the Imperial Valley MW 6.5 earthquake. The pulse-like ground motions are simulated by a kinematic approach using a fourth-order accurate finite-difference method. The results show that the slip asperity in the northern part of the fault plane contributes more to large velocity pulses than the asperity that originates in the hypocentral zone. The two-sided pulses on the fault-normal component and the one-sided pulses on the fault-parallel component are mainly regulated by the fault rupture and slip directions, respectively. Compared with the pseudo-velocity response spectrum, the displacement response spectrum better captures the resonance effect of long-period pulse-like ground motion on large structures. The location of the maximum ground motion coincides with the surface projection area of the large asperity, and the amplitude and the attenuation rate of each horizontal component vary with increasing distance from the fault. The difference in the ground motion peaks on both sides of the fault implies that the fault-normal component is more sensitive to the fault dip angle than the fault-parallel component. We reproduce the pulse-like ground motions up to 1 Hz by fitting the simulated velocity time histories, response spectrum and ground motions to observed data. Using the pulse records of the Imperial Valley earthquake, the 1994 Northridge MW 6.7 earthquake, and the 1999 Chi-Chi MW 7.6 earthquake, we compare the effects of earthquake magnitude on peak value and distribution of large velocity pulses, in which a reasonable consistency can be observed. Our simulated results of near-field pulse-like ground motions are beneficial to clarify the pulse mechanism.

A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

Abstract We analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on m ≥ 1 {m\geq 1} intervals of length k m {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form C 0 ⁢ h 2 + C m ⁢ k {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where C m ≤ C ′ + C ′′ m {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}} . This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

Fourth order compact scheme for space fractional advection–diffusion reaction equations with variable coefficients

In this work, a new fourth order compact approximation is derived for Riemann–Liouville space fractional derivatives. Modified wave numbers are obtained for various approximations of fractional derivatives. Considering these modified wave numbers, Fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations. A novel fourth order compact scheme is developed for space fractional advection–diffusion reaction equations with variable coefficients in one and two dimensions. In the proposed compact scheme, the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations. This splitting of second derivative approximations allows us to obtain a similar system of linear equations as in Zhao and Tian (2017) which presents only second order accurate finite difference method for advection–diffusion reaction equations in one-dimension. The stability of the proposed compact scheme is demonstrated numerically. Furthermore, the proposed compact scheme is employed to space fractional Black–Scholes equation for pricing European options as an application of fractional derivatives in mathematical finance. Numerical illustrations are presented to validate the theoretical claims.

Stable Symmetric Matrix Form Framework for the Elastic Wave Equation Combined with Perfectly Matched Layer and Discretized in the Curve Domain

In this paper, we present a stable and accurate high-order methodology for the symmetric matrix form (SMF) of the elastic wave equation. We use an accurate high-order upwind finite difference method to define spatial discretization. Then, an efficient complex frequency-shifted (CFS) unsplit multi-axis perfectly matched layer (MPML) is implemented using the auxiliary differential equation (ADE) that is used to build higher-order time schemes for elastodynamics in the unbounded curve domain. It is derived to be compatible with SMF. The SMF framework has a general form of a hyperbolic partial differential equation (PDE) that can be expanded to different dimensions (2D, 3D) or different wave modal (SH, P-SV) without requiring significant modifications owing to a simplified process of derivation and programming. Subsequently, an energy analysis on the framework combined with initial boundary value problems is conducted, and the stability analysis can be extended to a semi-discrete approximation similarly. Thus, we propose a semi-discrete approximation based on ADE CFS-MPML in which the curve domain is discretized using the upwind summation-by-parts (SBP) operators, and where the boundary conditions are enforced weakly using the simultaneous approximation terms (SAT). The proposed method’s robustness and adequacy are illustrated by conducting several numerical simulations.

A new high-resolution two-level implicit method based on non-polynomial spline in tension approximations for time-dependent quasi-linear biharmonic equations with engineering applications

In this present work, a new two-level implicit non-polynomial spline in tension method is proposed for the numerical solution of the 1D unsteady quasi-linear biharmonic problem. Using the continuity of the first-order derivative of the spline in tension function, a fourth-order accurate implicit finite-difference method is developed in this manuscript. By considering the linear model biharmonic problem, the given implicit spline method is unconditionally stable. Since the proposed method is based on half-step grid points, so it can be directly applied to 1D singular biharmonic problems without alteration in the scheme. Finally, the numerical experiments of the various biharmonic equation such as the generalized Kuramoto–Sivashinsky equation, extended Fisher–Kolmogorov equation, and 1D linear singular biharmonic problems are carried out to show the efficacy, accuracy, and reliability of the method. From the computational experiments, improved numerical results obtained as compared to the results obtained in earlier research work.

Numerical prediction of solidified shell thicknesses obtained in continuous casting with different billet shapes

Thermal physics of continuous casting is strongly dependent on the velocity of the molten metal, surface heat extraction rate and also on the mold shapes. In the present numerical study, both the moving front and the geometric boundaries of non-rectilinear molds are modeled using Immersed Boundary Method (IBM) which reduces computational time and provides simplicity as compared to the body confirming grid approaches. A second order accurate finite difference method is used to solve unsteady heat conduction equation along with lateral convection to predict the solidified shell thickness during continuous casting. This methodology shows good agreement with the reported data for a rectangular billet. A quantitative study has been performed to report the effects of casting velocity and heat extraction rate on the solidified shell thickness for different mold shapes such as circular, elliptic and I-shaped. Chipman-Fondersmith correlations for solidified shell thickness during continuous casting are assessed for these three different mold shapes and improved correlations are suggested to take into account the variations in the heat extraction rate.

Parabolic two-step model and accurate numerical scheme for nanoscale heat conduction induced by ultrashort-pulsed laser heating

Simulation of nanoscale heat transfer phenomena has attracted great attention, particularly the nano-scale heat conduction induced by ultrashort-pulsed laser heating. In this article, we propose a nanoscale parabolic two-step model and an accurate numerical scheme for thermal analysis of the nanoscale heat conduction induced by ultrashort-pulsed laser heating. To this end, we first introduce the Knudsen number (Kn) into the original parabolic two-step heat conduction equations and couple them with the Kn-dependent and temperature-jump boundary condition. We then develop a fourth-order accurate compact finite difference method for solving the nanoscale model. Stability and convergence of the obtained numerical scheme are analyzed theoretically. We finally test the accuracy and applicability of the nanoscale model and the obtained numerical scheme in three examples. By choosing various values of the Kn and the parameter α in the boundary condition, the simulation could be a tool for analyzing the nanoscale heat conduction induced by ultrashort-pulsed laser heating.

A fourth-order accurate finite difference method to evaluate the true transient behaviour of natural convection flow in enclosures

PurposeModelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving the higher order temporal accuracy. A fourth-order accurate finite difference method in both space and time is proposed to overcome these numerical errors and accurately model the transient behaviour of natural convection flow in enclosures using vorticity–streamfunction formulation.Design/methodology/approachFourth-order wide stencil formula with appropriate one-sided difference extrapolation technique near the boundary is used for spatial discretisation, and classical fourth-order Runge–Kutta scheme is applied for transient term discretisation. The proposed method is applied on two transient case studies, i.e. convection–diffusion of a Gaussian Pulse and Taylor Vortex flow having analytical solution.FindingsError magnitude comparison and rate of convergence analysis of the proposed method with these analytical solutions establish fourth-order accuracy and prove the ability of the proposed method to truly capture the transient behaviour of incompressible flow. Also, to test the transient natural convection flow behaviour, the algorithm is tested on differentially heated square cavity at high Rayleigh number in the range of 103-108, followed by studying the transient periodic behaviour in a differentially heated vertical cavity of aspect ratio 8:1. An excellent comparison is obtained with standard benchmark results.Research limitations/implicationsThe developed method is applied on 2D enclosures; however, the present methodology can be extended to 3D enclosures using velocity–vorticity formulations which shall be explored in future.Originality/valueThe proposed methodology to achieve fourth-order accurate transient simulation of natural convection flows is novel, to the best of the authors’ knowledge. Stable fourth-order vorticity boundary conditions are derived for boundary and external boundary regions. The selected case studies for comparison demonstrate not only the fourth-order accuracy but also the considerable reduction in error magnitude by increasing the temporal accuracy. Also, this study provides novel benchmark results at five different locations within the differentially heated vertical cavity of aspect ratio 8:1 for future comparison studies.

Numerical Approximations for the Tempered Fractional Laplacian: Error Analysis and Applications

In this paper, we propose an accurate finite difference method to discretize the $d$-dimensional (for $d\ge 1$) tempered integral fractional Laplacian and apply it to study the tempered effects on the solution of problems arising in various applications. Compared to other existing methods, our method has higher accuracy and simpler implementation. Our numerical method has an accuracy of $O(h^\epsilon)$, for $u \in C^{0, \alpha+\epsilon} (\bar{\Omega})$ if $\alpha < 1$ (or $u \in C^{1, \alpha-1+\epsilon} (\bar{\Omega})$ if $\alpha \ge 1$) with $\epsilon > 0$, suggesting the minimum consistency conditions. The accuracy can be improved to $O(h^2)$, for $u \in C^{2, \alpha+\epsilon} (\bar{\Omega})$ if $\alpha < 1$ (or $u \in C^{3, \alpha - 1 + \epsilon} (\bar{\Omega})$ if $\alpha \ge 1$). Numerical experiments confirm our analytical results and provide insights in solving the tempered fractional Poisson problem. It suggests that to achieve the second order of accuracy, our method only requires the solution $u \in C^{1,1}(\bar{\Omega})$ for any $0<\alpha<2$. Moreover, if the solution of tempered fractional Poisson problems satisfies $u \in C^{p, s}(\bar{\Omega})$ for $p = 0, 1$ and $0<s \le 1$, our method has the accuracy of $O(h^{p+s})$. Since our method yields a (multilevel) Toeplitz stiffness matrix, one can design fast algorithms via the fast Fourier transform for efficient simulations. Finally, we apply it together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including the Allen-Cahn equation and Gray-Scott equations.

A computational method for earthquake cycles within anisotropic media

SUMMARYWe present a numerical method for the simulation of earthquake cycles on a 1-D fault interface embedded in a 2-D homogeneous, anisotropic elastic solid. The fault is governed by an experimentally motivated friction law known as rate-and-state friction which furnishes a set of ordinary differential equations which couple the interface to the surrounding volume. Time enters the problem through the evolution of the ordinary differential equations along the fault and provides boundary conditions for the volume, which is governed by quasi-static elasticity. We develop a time-stepping method which accounts for the interface/volume coupling and requires solving an elliptic partial differential equation for the volume response at each time step. The 2-D volume is discretized with a second-order accurate finite difference method satisfying the summation-by-parts property, with boundary and fault interface conditions enforced weakly. This framework leads to a provably stable semi-discretization. To mimic slow tectonic loading, the remote side-boundaries are displaced at a slow rate, which eventually leads to earthquake nucleation at the fault. Time stepping is based on an adaptive, fourth-order Runge–Kutta method and captures the highly varying timescales present. The method is verified with convergence tests for both the orthotropic and fully anisotropic cases. An initial parameter study reveals regions of parameter space where the systems experience a bifurcation from period one to period two behaviour. Additionally, we find that anisotropy influences the recurrence interval between earthquakes, as well as the emergence of aseismic transients and the nucleation zone size and depth of earthquakes.

The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems

We propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of time-dependent diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. The method of lines (MOL) is used to combine spatial and temporal discretizations. The spatial scheme requires the definition of a high-order approximation of the divergence and gradient operators and the two inner products for the discrete analogs of fluxes and scalar unknowns. The discrete divergence and gradient operators are built according to a discrete duality relation. The inner product for the flux grid functions is built by explicitly imposing the conditions of consistency and stability. The family of semi-discrete mimetic methods is proved theoretically to be energy-stable as the corresponding continuous problem. Then, a full discretization is derived by combining via MOL the MFD method of order k with time marching schemes from the backward differentiation formula of order $$k+2$$ . Optimal order of accuracy is demonstrated for the scalar variable and verified numerically by solving the time-dependent diffusion problems with a variable diffusion tensor for k from 0 to 3 on three different unstructured mesh families.

Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications

In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian (−Δ)α2 (0<α<2) in hypersingular integral form. The proposed methods provide a fractional analogue of the central difference schemes to the fractional Laplacian. As α→2−, they collapse to the central difference schemes of the classical Laplace operator −Δ. We prove that our methods are consistent if u∈C⌊α⌋,α−⌊α⌋+ε(Rd), and the local truncation error is O(hε), with ε>0 a small constant and ⌊⋅⌋ denoting the floor function. If u∈C2+⌊α⌋,α−⌊α⌋+ε(Rd), they can achieve the second order of accuracy for any α∈(0,2). These results hold for any dimension d≥1 and thus improve the existing error estimates of the one-dimensional cases in the literature. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen–Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should at most satisfy u∈C1,1(Rd). One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two- and three-dimensional fractional Allen–Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems.

A compact high order Alternating Direction Implicit method for three-dimensional acoustic wave equation with variable coefficient

Efficient and accurate numerical simulation of seismic wave propagation is important in various Geophysical applications such as seismic full waveform inversion (FWI) problem. However, due to the large size of the physical domain and requirement on low numerical dispersion, many existing numerical methods are inefficient for numerical modelling of seismic wave propagation in an heterogeneous media. Despite the great efforts that have been devoted during the past decades, it still remains a challenging task in the development of efficient and accurate finite difference method for multi-dimensional acoustic wave equation with variable velocity. In this paper we proposed a Padé approximation based finite difference scheme for solving the acoustic wave equation in three-dimensional heterogeneous media. The new method is obtained by combining the Padé approximation and a novel algebraic manipulation. The efficiency of the new algorithm is further improved through the Alternative Directional Implicit (ADI) method. The stability of the new algorithm has been theoretically proved by energy method. The new method is conditionally stable with a better Courant–Friedrichs–Lewy condition (CFL) condition, which has been verified numerically. Extensive numerical examples have been solved, which demonstrated that the new method is accurate, efficient and stable.

Accurate gradient preserved method for solving heat conduction equations in double layers

Analyzing heat transfer in layered structures is important for the design and operation of devices and the optimization of thermal processing of materials. Existing numerical methods dealing with layered structures if using only three grid points across the interface usually provide only a second-order truncation error. Obtaining a numerical scheme using three grid points across the interface so that the overall numerical scheme is unconditionally stable and convergent with higher-order accuracy is mathematically challenging. In this study, we develop a higher-order accurate finite difference method using three grid points across the interface by preserving the first-order derivative, ux, in the interfacial condition and/or the boundary condition. As such, when the compact fourth-order accurate Padé scheme is used at the interior points, the overall scheme maintains to be higher-order accurate. Stability and convergence of the scheme are analyzed. Finally, four examples are given to test the obtained numerical method. Results show that the convergence order in space is close to 4.0, which coincides with the theoretical analysis.

Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere

Abstract In this article, the combined magnetohydrodynamic heat, momentum and mass (species) transfer in external boundary layer flow of Casson nanofluid from an isothermal sphere surface with convective condition under an applied magnetic field is studied theoretically. The effects of Brownian motion and thermophoresis are incorporated in the model in the presence of both heat and nanoparticle mass transfer convective conditions. The governing partial differential equations (PDEs) are transformed into highly nonlinear, coupled, multi-degree non-similar partial differential equations consisting of the momentum, energy and concentration equations via appropriate non-similarity transformations. These transformed conservation equations are solved subject to appropriate boundary conditions with a second order accurate finite difference method of the implicit type. The influences of the emerging parameters i.e. magnetic parameter (M), Buoyancy ratio parameter (N), Casson fluid parameter (β), Brownian motion parameter (Nb) and thermophoresis parameter (Nt), Lewis number (Le), Prandtl number (Pr) and thermal slip (ST) on velocity, temperature and nano-particle concentration distributions is illustrated graphically and interpreted at length. Increasing viscoplastic (Casson) parameter decelerates the flow and also decreases thermal and nano-particle concentration. Increasing Brownian motion accelerates the flow and enhances temperatures whereas it reduces nanoparticle concentration boundary layer thickness. Increasing thermophoretic parameter increasing momentum (hydrodynamic) boundary layer thickness and nanoparticle boundary layer thickness whereas it reduces thermal boundary layer thickness. Increasing magnetohydrodynamic body force parameter decelerates the flow whereas it enhances temperature and nano-particle (species) concentrations. The study is relevant to enrobing processes for electric-conductive nano-materials, of potential use in aerospace and other industries.

High-Order Accurate Conservative Finite Difference Methods for Vlasov Equations in 2D+2V

This manuscript discusses discretization of the Vlasov--Poisson system in 2D+2V phase space using high-order accurate conservative finite difference algorithms. One challenge confronting direct kinetic simulation is the significant computational cost associated with the high-dimensional phase space description. In the present work we advocate the use of high-order accurate schemes as a mechanism to reduce the computational cost required to deliver a given level of error in the computed solution. We pursue a discretely conservative finite difference formulation of the governing equations, and discuss fourth- and sixth-order accurate schemes. In addition, we employ a minimally dissipative nonlinear scheme based on the well-known WENO (weighted essentially nonoscillatory) approach. Verification of the full formulation is performed using the method of manufactured solutions. Results are also presented for the physically relevant scenarios of Landau damping, and growth of transverse instabilities from an imposed plane wave.

A Fully Fourth Order Accurate Energy Stable Finite Difference Method for Maxwell's Equations in Metamaterials

We present a novel fully fourth order in time and space finite difference method (FDM) for the time domain Maxwell's equations in metamaterials. We consider a Drude metamaterial model for the material response to incident electromagnetic fields. We consider the second-order formulation of the system of partial differential equations that govern the evolution in time of electric and magnetic fields along with the evolution of the polarization and magnetization current densities. Our discretization employs fourth-order staggering in space of different field components and the modified equation approach to obtain fourth-order accuracy in time. Using the energy method, we derive energy relations for the continuous models and design numerical schemes that preserve a discrete analogue of the energy relation. Numerical simulations are provided in one- and two-dimensional settings to illustrate fourth-order convergence as well as to compare with second-order schemes.