Stable Finite Difference Method Research Articles

High-order entropy stable schemes based on a local adaptive WENO viscosity for degenerate convection-diffusion equations

In this work, high-order entropy stable finite difference methods are constructed to solve systems of degenerate convection-diffusion partial differential equations. The approximation of the convective term is based on high-order entropy stable fluxes under a TeCNO scheme formulation and for the diffusive term high-order entropy conservative fluxes are computed by considering linear combinations of second-order approximations. Numerical simulations show that the proposed high-order entropy conservative schemes provide a convergence rate better than conventional WENO methods. To obtain entropy stability, a local adaptive artificial viscosity is included using weighted essentially nonoscillatory reconstructions satisfying a local sign property at discontinuities. Different WENO interpolations for the local artificial viscosity are numerically analyzed being the fast and optimal WENO interpolation (FOWENO), the one that introduces less spurious oscillations with a lower computational cost.

Two fast and unconditionally stable finite difference methods for Riesz fractional diffusion equations with variable coefficients

In this paper, for variable coefficient Riesz fractional diffusion equations in one and two dimensions, we first design a second-order implicit difference scheme by using the Crank-Nicolson method and a fractional centered difference formula for time and space variables, respectively. With the compact operator acting on, a novel fourth-order finite difference scheme is subsequently constructed. Solvability, stability and convergence of these schemes are theoretically analyzed. For these discretized linear systems, the fast implementation with preconditioners based on sine transform is proposed, which has the computational complexity of O(nlog⁡n) per iteration and the memory requirement of O(n), where n represents the total number of the spatial grid nodes. Finally, numerical experiments are performed to illustrate the preciseness and effectiveness of these new techniques.

A Stable Finite Difference Method Based on Upward Continuation to Evaluate Vertical Derivatives of Potential Field Data

A Stable Finite Difference Method Based on Upward Continuation to Evaluate Vertical Derivatives of Potential Field Data

An explicit stable finite difference method for the Allen–Cahn equation

We propose an explicit stable finite difference method (FDM) for the Allen–Cahn (AC) equation. The AC equation has been widely used for modeling various phenomena such as mean curvature flow, image processing, crystal growth, interfacial dynamics in material science, and so on. For practical use, an explicit method can be applied for the numerical approximation of the AC equation. However, there is a strict restriction on the time step size. To mitigate the disadvantage, we adopt the alternating direction explicit method for the diffusion term of the AC equation. As a result, we can use a relatively larger time step size than when the explicit method is used. Numerical experiments are performed to demonstrate that the proposed scheme preserves the intrinsic properties of the AC equation and it is stable compared to the explicit method.

Finite difference methods with linear interpolation for solving a coupled system of hyperbolic delay differential equations

In this article a system of first-order hyperbolic delay differential equations is considered. The maximum principle is proved for the problem considered. Further, the stability of the solution is established as an application of the maximum principle. The propagation of the discontinuity of the solution is also established. A piecewise uniform mesh is designed for solving the problem. On the piecewise uniform mesh, conditionally stable and unconditionally stable finite difference methods with piecewise linear interpolation are suggested to solve the problem. It is proved that the methods are consistent, stable, and convergent. Numerical illustrative examples are given to validate the theoretical results.

An energy stable finite difference method for anisotropic surface diffusion on closed curves

In this paper, we develop an energy stable finite difference method for the problem of curve motion under anisotropic surface diffusion. The motion of curve by anisotropic surface diffusion is governed by the fourth-order (highly nonlinear) geometric evolution equation. As in Li and Bao (2021), we first split the fourth-order evolution equation into two second-order equations, where the position vector of curve and the weighted curvature are treated as unknowns. Instead of using the arclength parameter, we introduce a Lagrangian coordinate parameter such that a closed curve can be parametrized over a fixed interval so that the equations can be represented using the new parameter. We then propose a linearly semi-implicit finite difference method to discretize these two equations, and prove that the scheme satisfies discrete energy dissipation so it is energy stable under suitable condition on the anisotropic surface energy. To show the applicability of our present method, we perform several numerical tests on various initial curves and different anisotropic energies. The numerical results show that our scheme is indeed energy dissipative and conserves even the total area well.

Drug release from viscoelastic polymeric matrices - a stable and supraconvergent FDM

Drug release from viscoelastic polymeric matrices is a complex phenomenon where the main actors are the fluid, the polymeric structure and the drug. As the fluid enters into the polymer, the polymeric chains relax inducing a resistance to the fluid entrance. In contact with the fluid, a dissolution processes takes place and the dissolved drug diffuses through the medium. Our main goal in this paper is to propose a stable finite difference method that leads to supraconvergent approximations for the fluid, solid and dissolved drug concentrations. The analysis proposed is non-standard in the sense that the error estimates have an important role in the stability analysis.

Stable finite difference method for fractional reaction\u2013diffusion equations by compact implicit integration factor methods

In this paper we propose a stable finite difference method to solve the fractional reaction–diffusion systems in a two-dimensional domain. The space discretization is implemented by the weighted shifted Grünwald difference (WSGD) which results in a stiff system of nonlinear ordinary differential equations (ODEs). This system of ordinary differential equations is solved by an efficient compact implicit integration factor (cIIF) method. The stability of the second order cIIF scheme is proved in the discrete L^{2}-norm. We also prove the second-order convergence of the proposed scheme. Numerical examples are given to demonstrate the accuracy, efficiency, and robustness of the method.

An unconditionally stable finite-difference method for the solution of multi-dimensional transport equation

The rightward representation of the Barakat-Clark ADE scheme is extended for the solution of the Multi-dimensional Transport equation. The first-order derivative of the Transport equation is represented by a one-sided multi-level finite-difference. The resulting scheme is an explicit marching type iterative solution. The visibility of using this method for the solution of the Multi-dimensional Transport equations is demonstrated through the solution of each of the Burgers’ equation and the Graetz-Nusselt problem for the thermally developing flow between two-parallel plates at constant temperature (at high and low Peclet numbers) with parabolic velocity distribution. The results are compared with the solutions using other schemes. All of the obtained results are compared with the exact solutions of the analyzed problems. The results show that the proposed scheme is unconditionally stable, better accuracy, faster convergence, and lower storage capacity requirement.

Stable Finite-Difference Methods for Elastic Wave Modeling with Characteristic Boundary Conditions

In this paper, a new stable finite-difference (FD) method for solving elastodynamic equations is presented and applied on the Biot and Biot/squirt (BISQ) models. This method is based on the operator splitting theory and makes use of the characteristic boundary conditions to confirm the overall stability which is demonstrated with the energy method. Through the stability analysis, it is showed that the stability conditions are more generous than that of the traditional algorithms. It allows us to use the larger time step τ in the procedures for the elastic wave field solutions. This context also provides and compares the computational results from the stable Biot and unstable BISQ models. The comparisons show that this FD method can apply a new numerical technique to detect the stability of the seismic wave propagation theories. The rigorous theoretical stability analysis with the energy method is presented and the stable/unstable performance with the numerical solutions is also revealed. The truncation errors and the detailed stability conditions of the FD methods with different characteristic boundary conditions have also been evaluated. Several applications of the constructed FD methods are presented. When the stable FD methods to the elastic wave models are applied, an initial stability test can be established. Further work is still necessary to improve the accuracy of the method.

Numerical solution of time‐fractional singularly perturbed convection–diffusion problems with a delay in time

This work is concerned with the development of a stable finite difference method (SFDM) for time‐fractional singularly perturbed convection–diffusion problems with a delay in time. The fractional derivative is considered in the Caputo sense. The SFDM is constructed based on the stability of the analytical solution. Unlike the other classical numerical methods for singular perturbation problems, this method works nicely on a uniform mesh. The method is easily adaptable on Shishkin mesh and also on any graded mesh in space. Error estimates are presented to show the convergence of the numerical scheme. To support the theory, numerical results are presented in tables for different values of the fractional derivative parameter and perturbation parameter.

Viscous dissipative free convective flow from a vertical cone with heat generation/absorption, MHD in the presence of radiative non-uniform surface heat flux

The problem of combined effects of heat generation/absorption and thermal radiation on unsteady, laminar, natural convective flow with MHD, viscous dissipation from a vertical cone in the presence of variable surface heat flux is considered here. The non-dimensional governing equations of the flow that is transient, coupled and non-linear PDE’s are solved by an efficient, accurate and unconditionally stable finite difference method of Crank-Nicolson type. Numerical results of the velocity and thermal profiles are obtained and analyzed to expose the results of different values of parameters Pr, M, n and Rd. The present results are compared with available results in literature and are found to be in excellent conformity. The study finds important applications in hybrid solar energy systems, geophysical heat transfer and industrial manufacturing processes.

Numerical Solutions of Fractional Differential Equations Arising in Engineering Sciences

This paper deals with the numerical solutions of a class of fractional mathematical models arising in engineering sciences governed by time-fractional advection-diffusion-reaction (TF–ADR) equations, involving the Caputo derivative. In particular, we are interested in the models that link chemical and hydrodynamic processes. The aim of this paper is to propose a simple and robust implicit unconditionally stable finite difference method for solving the TF–ADR equations. The numerical results show that the proposed method is efficient, reliable and easy to implement from a computational viewpoint and can be employed for engineering sciences problems.

A one-dimensional mathematical simulation to salinity control in a river with a barrage dam using an unconditionally stable explicit finite difference method

Salinity refers to the amount of salt in rivers, where the salt can be in many different forms. There are two main methods of defining the concentration of salt in water such as the total dissolved solid measurement (TDS) and the electrical conductivity measurement (EC). The salinity is measured by evaporating water to dryness and weighing the solid residue. The electrical conductivity measurement is measured by passing an electric current through the water and measuring how readily the current flows. The total amount of salt in the water can affect the taste of water. The World Health Organization’s guideline on water palatability is that water with a salinity level of less than about 0.50–0.60 g/L is generally considered to be of a standard level. The drinking-water becomes significantly and increasingly unpalatable at salinity levels greater than about 1.0 g/L. In this research, a one-dimensional mathematical model of salinity measurement in a river is proposed. A modified model of salinity control in a river with a barrage dam is also introduced. An unconditionally stable explicit finite difference technique is used to approximate the salinity level under several conditions from the proposed model. The proposed computational technique gives good agreement results in realistic scenarios for water supply processes.

A Non-dimensional Mathematical Model of Salinity Measurement in the Chaophraya River Using an Unconditionally Stable Finite Difference Method

A Non-dimensional Mathematical Model of Salinity Measurement in the Chaophraya River Using an Unconditionally Stable Finite Difference Method

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.

Alternating-direction implicit finite difference methods for a new two-dimensional two-sided space-fractional diffusion equation

According to the principle of conservation of mass and the fractional Fick’s law, a new two-sided space-fractional diffusion equation was obtained. In this paper, we present two accurate and efficient numerical methods to solve this equation. First we discuss the alternating-direction finite difference method with an implicit Euler method (ADI–implicit Euler method) to obtain an unconditionally stable first-order accurate finite difference method. Second, the other numerical method combines the ADI with a Crank–Nicolson method (ADI–CN method) and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method. Finally, numerical solutions of two examples demonstrate the effectiveness of the theoretical analysis.

Investigations on several high-order ADI methods for time-space fractional diffusion equation

The paper is devoted to the construction of high-precision unconditionally stable finite difference methods for solving time-space fractional diffusion equation with the Caputo fractional derivative (of order β, with β ∈ (0, 1)) in time and the Rimann-Liouville fractional derivatives (of order α, with α ∈ (1, 2]) in space. Two kinds of difference schemes with the approximation orders O(τ2−β + h3) and O(τ2 + h3) respectively are constructed. The stability and convergence are analyzed in detail. The obtained results are illustrated numerically by some examples, and a comparative study of several high-order schemes is also carried out.

Numerical solution of the electron transport equation in the upper atmosphere

A new approach for solving the electron transport equation in the upper atmosphere is derived. The problem is a very stiff boundary value problem, and to obtain an accurate numerical solution, matrix factorizations are used to decouple the fast and slow modes. A stable finite difference method is applied to each mode. This solver is applied to a simplified problem for which an exact solution exists using various versions of the boundary conditions that might arise in a natural auroral display. The numerical and exact solutions are found to agree with each other, verifying the method.

Simulation of acoustic and flexural-gravity waves in ice-covered oceans

We introduce a provably stable, high-order-accurate finite difference method for simulation of acoustic and flexural gravity waves in compressible, inviscid fluids partially covered by a thin elastic layer. Such waves arise when studying ocean wave interactions with floating ice shelves, sea ice, and floating structures. Particular emphasis is on a well-posed interface treatment of the fluid-ice coupling. To ensure numerical stability and efficiency, finite difference approximations based on the summation-by-parts (SBP) framework are combined with a penalty technique (simultaneous approximation term, SAT) to impose the boundary and interface conditions. The resulting SBP-SAT approximations are time integrated with an unconditionally stable finite difference method. Numerical simulations in 2D corroborate the predicted efficiency and stability behaviors. The method can be used in its current form to study transmission of ocean waves and tsunamis through ice shelves, and upon coupling to an elastic half-space beneath the ice and water, to study ice motions associated with long-period seismic surface waves.