Accurate Finite Difference Scheme Research Articles

A High Accuracy Local One-Dimensional Explicit Compact Scheme for the 2D Acoustic Wave Equation

In this paper, we develop a highly accurate and efficient finite difference scheme for solving the two-dimensional (2D) wave equation. Based on the local one-dimensional (LOD) method and Padé difference approximation, a fourth-order accuracy explicit compact difference scheme is proposed. Then, the Fourier analysis method is used to analyze the stability of the scheme, which shows that the new scheme is conditionally stable and the Courant-Friedrichs-Lewy (CFL) condition is superior to most existing methods of equivalent order of accuracy in the literature. Finally, numerical experiments demonstrate the high accuracy, stability, and efficiency of the proposed method.

Granular Flow–Obstacle Interaction and Granular Dam Break Using the S-H Model with the TVD-MacCormack Scheme

An accurate second-order spatial and temporal finite-difference scheme is applied to solve the dynamics model of a depth-averaged avalanche. Within the framework of the MacCormack scheme, a total variation diminishing term supplements the corrector step to suppress large oscillations in domains with steep gradients. The greatest strength of the scheme lies in its high computational efficiency while maintaining satisfactory accuracy. The performance of the scheme is tested on a granular flume flow–obstacle interaction scenario and a granular dam breaking scenario. In the former, the flume flow splits into two granular streams when an obstacle is encountered. The opening between the two granular streams widens when the side length of the obstacle increases. In the simulation, shock waves with a fan-shaped configuration are captured, and successive waves in the tail of the avalanche between the two streams are observed. In the latter scenario, the average values and the fluctuations in the flow rate and velocity (at relatively steady state) decrease with the width of the breach. The capture of complex and typical granular-flow phenomena indicates the applicability and effectiveness of combining the TVD-MacCormack Scheme and S-H model to simulate dam breaking and inclined flow–obstacle interaction cases. In this study, the dense granular flow strikes on a rigid obstacle that is described by a wall boundary, rather than a topographic feature with a finite slope. This shows that the TVD-MacCormack scheme has a shock-capturing ability. The results of granular dam break simulations also revealed that the boundary conditions (open or closed) affect the collapse of the granular pile, i.e., the grains evenly breached out under closed boundary conditions, whereas the granules breaching out of the opening were mostly grains adjacent to the boundaries under open boundary conditions.

A Fifth-Order-Accurate Finite Difference Scheme for a Natural Coordinate System with Non-Uniform Grid

A fifth-order accurate finite difference scheme for the first derivative in von Mises coordinates is developed in this work. The scheme is tested in the computation of boundary vorticity in the study of two-dimensional flow of a viscous fluid in a curvilinear channel. Results obtained show an improvement in the computed solution over fourth-order accurate scheme

High-order accurate schemes for Maxwell's equations with nonlinear active media and material interfaces

We describe a fourth-order accurate finite-difference time-domain scheme for solving dispersive Maxwell's equations with nonlinear multi-level carrier kinetics models. The scheme is based on an efficient single-step three time-level modified equation approach for Maxwell's equations in second-order form for the electric field coupled to ODEs for the polarization vectors and population densities of the atomic levels. The resulting scheme has a large CFL-one time-step. Curved interfaces between different materials are accurately treated with curvilinear grids and compatibility conditions. A novel hierarchical modified equation approach leads to an explicit scheme that does not require any nonlinear iterations. The hierarchical approach at interfaces leads to local updates at the interface with no coupling in the tangential directions. Complex geometry is treated with overset grids. Numerical stability is maintained using high-order upwind dissipation designed for Maxwell's equations in second-order form. The scheme is carefully verified for a number of two and three-dimensional problems. The resulting numerical model with generalized dispersion and arbitrary nonlinear multi-level system can be used for many plasmonic applications such as for ab initio time domain modeling of nonlinear engineered materials for nanolasing applications, where nano-patterned plasmonic dispersive arrays are used to enhance otherwise weak nonlinearity in the active media.

High-order accurate entropy stable adaptive moving mesh finite difference schemes for special relativistic (magneto)hydrodynamics

This paper develops high-order accurate entropy stable (ES) adaptive moving mesh finite difference schemes for the two- and three-dimensional special relativistic hydrodynamic (RHD) and magnetohydrodynamic (RMHD) equations, which is the high-order accurate extension of Duan and Tang (2021) [23]. The key point is the derivation of the higher-order accurate entropy conservative (EC) and ES finite difference schemes in the curvilinear coordinates by carefully dealing with the discretization of the temporal and spatial metrics and the Jacobian of the coordinate transformation and constructing the high-order EC and ES fluxes with the discrete metrics. The spatial derivatives in the source terms of the symmetrizable RMHD equations and the geometric conservation laws are discretized by using the linear combinations of the corresponding second-order case to obtain high-order accuracy. Based on the proposed high-order accurate EC schemes and the high-order accurate dissipation terms built on the WENO reconstruction, the high-order accurate ES schemes are obtained for the RHD and RMHD equations in the curvilinear coordinates. The mesh iteration redistribution or adaptive moving mesh strategy is built on the minimization of the mesh adaption functional. Several numerical tests are conducted to validate the shock-capturing ability and high efficiency of our high-order accurate ES adaptive moving mesh methods on the parallel computer system with the MPI communication. The numerical results show that the high-order accurate ES adaptive moving mesh schemes outperform both their counterparts on the uniform mesh and the second-order ES adaptive moving mesh schemes.

A new absorbing layer for simulation of wave propagation based on a KdV model on unbounded domain

Simulation for wave propagation based on the Korteweg-de Vries (KdV) equation on unbounded domain requires the computational domain to be bounded and hence an absorbing boundary condition is needed so that when the wave arrives the boundary it will not be reflected and destroy the simulation. In this article, we present a new absorbing (or called lossy) layer for simulation of wave propagation based on a KdV model on unbounded domain with non-homogeneous boundary condition at one end, and propose a two-level second-order accurate finite difference scheme for solving this KdV problem. We prove that both the analytical solution and the numerical solution are stable in L2-norm and l2-norm, respectively, and they decay in the lossy layer. Numerical examples are given to illustrate the new method and verify its effectiveness.

An accurate, robust, and efficient finite difference scheme with graded meshes for the time-fractional Burgers’ equation

This work presents the L1 implicit difference scheme based on non-uniform meshes for the time-fractional Burgers equation. We find that the difficulty arising from the singularity of the exact solution at t=0 can be overcome by non-uniform meshes. We yield the unconditional stability and optimal convergence rate of h2+τmin{rα,2−α} by means of the energy method, when selecting r=(2−α)/α. Convergence rates are higher than some recently studied schemes. Numerical experiments confirm the predicted theoretical estimate.

Discrete maximum principle of a high order finite difference scheme for a generalized Allen–Cahn equation

We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation of the $Q^2$ spectral element method. We prove that the discrete maximum principle holds under suitable mesh size and time step constraints. The same result also applies to construct a bound-preserving scheme for any passive convection with an incompressible velocity field.

Numerical analysis of a high-order accurate compact finite difference scheme for the SRLW equation

In this paper, we develop a fourth-order accurate compact difference scheme for the symmetric regularized long wave (SRLW) equation for a single nonlinear velocity form. The discrete conservation, priori estimate, solvability, convergence and stability of the present scheme are proved by the discrete energy method. Numerical examples are given to support the theoretical analysis.

Diffusion of dermatological irritant in drying laundered cloth.

Sodium dodecyl sulphate (SDS), a commonly used laundry surfactant, has been known to cause some damage to epithelial cells in skin. Further, independent experiments have shown that a single laundry wash with rinsing leaves a residue of around 10% of the chemicals used in a wash cycle. A realistic nonlinear system of partial differential equations is developed for coupled water and solute transport through a drying porous medium when the solute has a mobile state (monomers) as well as an immobile state (micelles). An accurate finite difference scheme is developed and tested against known exact solutions of the nonlinear porous medium equation for transport of water and against known conservation laws. It shows that at the end of atmosphere-controlled stage 1 of drying when little water remains, the concentration of SDS near the drying surface, where it may contact skin, is commonly an order of magnitude higher than its initial value. The problem is exacerbated by successive regular wash cycles and by higher evaporation rates in electronic dryers. The numerical solutions show the partitioning between the two phases of SDS.

Coupled and decoupled high‐order accurate dissipative finite difference schemes for the dissipative generalized symmetric regularized long wave equations

AbstractIn this paper, two coupled and decoupled dissipative finite difference schemes with high‐order accuracy are proposed for solving the dissipative generalized symmetric regularized long wave equations. Dissipation of the discrete energy of scheme with different parameters is discussed. A priori estimate, existence and uniqueness of numerical solutions, convergence with and stability of the schemes are proved by the discrete energy method. Numerical examples are given to support the theoretical analysis.

A new high‐order accurate conservative finite difference scheme for the coupled nonlinear Schrödinger equations

In this paper, a fourth‐order accurate conservative finite difference scheme for solving the coupled nonlinear Schrödinger (CNLS) equations is proposed. Conservation of the discrete energy and masses, priori estimates, existence and uniqueness of numerical solutions, convergence with second‐order in time and fourth‐order in space, and stability of the present scheme are proved by the discrete energy method. A convergent iterative method for the present scheme is developed. Numerical experiments are given to support the theoretical analysis.

A comprehensive study of secondary and tertiary vortex phenomena of flow past a circular cylinder: A Cartesian grid approach

This study envisages undertaking a comprehensive simulation of the flow past an impulsively started circular cylinder with special emphasis on vortex dynamics in the secondary and tertiary level. A recently developed second order spatially and temporally accurate compact finite difference scheme on a nonuniform Cartesian grid has been used to discretize the transient Navier–Stokes equations governing the flow. The grid is generated in such a way that the cylinder boundary passes through the grid points, thus dispensing with the need to use the immersed interface approach on a Cartesian grid. High quality simulations are accomplished for a wide range of Reynolds numbers (Re) from 5≤Re≤10 000 in the laminar regime, including the periodic flow characterized by von Kármán vortex street. The α, β, sub-α, and sub-β phenomena, which are the trademark of the secondary and tertiary vortex dynamics associated with such flows, are studied in detail. Our results are compared with existing experimental and numerical results, and close comparison is obtained in all the cases exemplifying their accuracy. In the process, for the first time, we also provide a tabular documentation of the early stages of the flow for Re≥700.

High-fidelity simulations of gravity currents using a high-order finite-difference spectral vanishing viscosity approach

This numerical work investigates the potential of a high-order finite-difference spectral vanishing viscosity approach to simulate gravity currents at high Reynolds numbers. The method introduces targeted numerical dissipation at small scales through altering the discretisation of the second derivatives of the viscous terms in the incompressible Navier-Stokes equations to mimic the spectral vanishing viscosity (SVV) operator, originally designed for the regularisation of spectral element method (SEM) solutions of pure advection problems. Using a sixth-order accurate finite-difference scheme, the adoption of the SVV method is straightforward and comes with a negligible additional computational cost. In order to assess the ability of this high-order finite-difference spectral vanishing viscosity approach, we performed large-eddy simulations (LES) of a gravity current in a channelised lock-exchange set-up with our SVV model and with the well-known explicit static and dynamic Smagorinsky sub-grid scale (SGS) models. The obtained data are compared with a direct numerical simulation (DNS) based on more than 800 million mesh nodes, and with experimental measurements. A framework for the energy budget is introduced to investigate the behaviour of the gravity current. First, it is found that the DNS is in good agreement with the experimental data for the evolution of the front location and velocity field as well as for the stirring and mixing inside the gravity current. Secondly, the LES performed with less than 0.4% of the total number of mesh nodes compared to the DNS, can reproduce the main features of the gravity currents, with the SVV model yielding slightly more accurate results. It is also found that the dynamic Smagorinsky model performs better than its static version. For the present study, the static and dynamic Smagorinsky models are 1.8 and 2.5 times more expensive than the SVV model, because the latter does not require the calculation of explicit SGS terms in the Navier-Stokes equations nor spatial filtering operations.

Triple Decomposition of Velocity Gradient Tensor in Compressible Turbulence

The decomposition of the local motion of a fluid into straining, shearing, and rigid-body rotation is examined in this work for a compressible isotropic turbulence by means of direct numerical simulations. The triple decomposition is closely associated with a basic reference frame (BRF), in which the extraction of the biasing effect of shear is maximized. In this study, a new computational and inexpensive procedure is proposed to identify the BRF for a three-dimensional flow field. In addition, the influence of compressibility effects on some statistical properties of the turbulent structures is addressed. The direct numerical simulations are carried out with a Reynolds number that is based on the Taylor micro-scale of Reλ=100 for various turbulent Mach numbers that range from Mat=0.12 to Mat=0.89. The DNS database is generated with an improved seventh-order accurate weighted essentially non-oscillatory scheme to discretize the non-linear advective terms, and an eighth-order accurate centered finite difference scheme is retained for the diffusive terms. One of the major findings of this analysis is that regions featuring strong rigid-body rotations or straining motions are highly spatially intermittent, while most of the flow regions exhibit moderately strong shearing motions in the absence of rigid-body rotations and straining motions. The majority of compressibility effects can be estimated if the scaling laws in the case of compressible turbulence are rescaled by only considering the solenoidal contributions.

Stability analysis of a high-order finite-difference scheme for the Korteweg–de Vries equation with non-homogeneous boundaries

In this paper, we analyze the stability of a high-order finite-difference scheme for the Korteweg–de Vries (KdV) equation with non-homogeneous boundaries. We first employ a variable transformation to change the non-homogeneous boundaries to homogeneous boundaries. We then develop a fourth-order accurate finite-difference scheme for solving the transformed KdV problem. The stability, convergence and solvability of the numerical solution are analyzed. Numerical examples are given to confirm the good accuracy and the effectiveness of the present method for handling the KdV equations with non-homogeneous boundaries.

Effects of Variable Thermal Conductivity and Grashof Number on Non-Darcian Natural Convection Flow of Viscoelastic Fluids with Non Linear Radiation and Dissipations

Effects of variable thermal conductivity, porosity and Grashof number on natural convection of viscoelastic fluid flow and heat transfer are studied and discussed. The nonlinear radiation and dissipations are taken into consideration. The fluid flows through non-Darcy porous medium which lies between two heated vertical plates that are kept at constant, but different, temperatures. High order accurate finite difference schemes are introduced to solve the governing equations. The coupled nonlinear differential equations are linearized and iterations are used to approximate that linearized terms. The finite difference method transforms the coupled linearized differential (momentum and energy) equations to a linear system of algebraic equations. An error analysis is introduced by refinement of mesh size and comparisons with available previous results. Effects of fluid and heat parameters on the velocity field, temperature, skin friction factor and Nusselt number are tabulated and plotted. The present results and comparisons show that the numerical solution is of excellent agreement with previous analytical and numerical solution.

High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics

This paper develops the high-order accurate entropy stable (ES) finite difference schemes for the shallow water magnetohydrodynamic (SWMHD) equations. They are built on the numerical approximation of the modified SWMHD equations with the Janhunen source term. First, the second-order accurate well-balanced semi-discrete entropy conservative (EC) schemes are constructed, satisfying the entropy identity for the given convex entropy function and preserving the steady states of the lake at rest (with zero magnetic field). The key is to match both discretizations for the fluxes and the non-flat river bed bottom and Janhunen source terms, and to find the affordable EC fluxes of the second-order EC schemes. Next, by using the second-order EC schemes as building block, high-order accurate well-balanced semi-discrete EC schemes are proposed. Then, the high-order accurate well-balanced semi-discrete ES schemes are derived by adding a suitable dissipation term to the EC scheme with the WENO reconstruction of the scaled entropy variables in order to suppress the numerical oscillations of the EC schemes. After that, the semi-discrete schemes are integrated in time by using the high-order strong stability preserving explicit Runge-Kutta schemes to obtain the fully-discrete high-order well-balanced schemes. The ES property of the Lax-Friedrichs flux is also proved and then the positivity-preserving ES schemes are studied by using the positivity-preserving flux limiters. Finally, extensive numerical tests are conducted to validate the accuracy, the well-balanced, ES and positivity-preserving properties, and the ability to capture discontinuities of our schemes.

Transportation of magnetite nanofluid flow and heat transfer over a rotating porous disk with Arrhenius activation energy: Fourth order Noumerov’s method

This article explains the effect of magnetite nanofluid taking into account water as the base fluid, over a rotating disk in the presence of external magnetic field. The governing boundary layer equations of laminar and incompressible flow over rotating disk have been formulated under the simultaneous influence of thermophoresis and Brownian motion. Additionally, influence of chemical reaction between species, prompted by Arrhenius activation energy is considered into the model. The Von Karman’s transformations of the Navier-Stokes equations are used to obtain the non-dimensional forms of the governing equations, and then fourth order finite difference scheme is used to find the numerical solutions using Noumerov’s discretization. This technique has strong impact on the accuracy of finite difference scheme. The physical descriptions of fluid velocity, temperature, concentration profiles and quantities of practical interest such as skin friction coefficient, Nusselt number and Sherwood number are presented with the help of graphs. One of the significant findings of this analysis include that an intensification in the thermophoresis parameter causes a downfall in the rate of heat transfer at the surface of the disk whereas another significant outcomes of the present investigation is that there is a downfall in the behaviours of the species concentration with the strengthening of Arrhenius activation energy. The fluid flow due to rotating disks has application in mechanisms of thermal transformations for nuclear propulsion devices.

Difference Scheme for the Numerical Solution of the Burgers Equation

A second-order accurate finite-difference scheme based on existing methods is proposed for the numerical solution of the one-dimensional Burgers equation. A stability condition is given under which the integration time step does not depend on the value of the viscous term. The numerical results produced by the scheme are compared with the exact solution of the Burgers equation.