Stable Finite Difference Scheme Research Articles

Why Stable Finite-Difference Schemes Can Converge to Different Solutions: Analysis for the Generalized Hopf Equation

The example of two families of finite-difference schemes shows that, in general, the numerical solution of the Riemann problem for the generalized Hopf equation depends on the finite-difference scheme. The numerical solution may differ both quantitatively and qualitatively. The reason for this is the nonuniqueness of the solution to the Riemann problem for the generalized Hopf equation. The numerical solution is unique in the case of a flow function with two inflection points if artificial dissipation and dispersion are introduced, i.e., the generalized Korteweg–de Vries-Burgers equation is considered. We propose a method for selecting coefficients of dissipation and dispersion. The method makes it possible to obtain a physically justified unique numerical solution. This solution is independent of the difference scheme.

Linear and Nonlinear Splitting Schemes Conserving Total Energy and Mass in the Shallow Water Model

Two linear and one nonlinear implicit unconditionally stable finite-difference schemes of the second-order approximation in all variables are given for a shallow-water model including the rotation and topography of the earth. The schemes are based on splitting the model equation into two one-dimensional subsystems. Each of the subsystems conserves the mass and total energy in both differential and discrete (in time and space) forms. One of the linear schemes contains a smoothing procedure not violating the conservation laws and suppressing spurious oscillations caused by the application of central-difference approximations of spatial derivatives. The unique solvability of the linear schemes and convergence of iterations used to find their solutions are proved.

Asymptotic Preserving and Uniformly Unconditionally Stable Finite Difference Schemes for Kinetic Transport Equations

Asymptotic Preserving and Uniformly Unconditionally Stable Finite Difference Schemes for Kinetic Transport Equations

Tamed stability of finite difference schemes for the transport equation on the half-line

In this paper, we prove that, under precise spectral assumptions, some finite difference approximations of scalar leftgoing transport equations on the positive half-line with numerical boundary conditions are ℓ 1 \ell ^1 -stable but ℓ q \ell ^q -unstable for any q > 1 q>1 . The proof relies on the accurate description of the Green’s function for a particular family of finite rank perturbations of Toeplitz operators whose essential spectrum belongs to the closed unit disk and with a simple eigenvalue of modulus 1 1 embedded into the essential spectrum.

A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws

The hyperbolic problem has a unique entropy solution, which maintains the entropy inequality. As such, people hope that the numerical results should maintain the discrete entropy inequalities accordingly. In view of this, people tend to construct entropy stable (ES) schemes. However, traditional numerical schemes cannot directly maintain discrete entropy inequalities. To address this, we here construct an ES finite difference scheme for the nonlinear hyperbolic systems of conservation laws. The proposed scheme can not only maintain the discrete entropy inequality, but also enjoy high-order accuracy. Firstly, we construct the second-order accurate semi-discrete entropy conservative (EC) schemes and ensure that the schemes meet the entropy identity when an entropy pair is given. Then, the second-order EC schemes are used as a building block to achieve the high-order accurate semi-discrete EC schemes. Thirdly, we add a dissipation term to the above schemes to obtain the high-order ES schemes. The term is based on the Weighted Essentially Non-Oscillatory (WENO) reconstruction. Finally, we integrate the scheme using the third-order Runge–Kutta (RK) approach in time. In the end, plentiful one- and two-dimensional examples are implemented to validate the capability of the scheme. In summary, the current scheme has sharp discontinuity transitions and keeps the genuine high-order accuracy for smooth solutions. Compared to the standard WENO schemes, the current scheme can achieve higher resolution.

Variation of Thermal Conductivity on Two-Dimensional Hydromagnetic Flow of Thermophoretic Forces: Impact of Suction/Blowing

The novelty of thermal conductivity is the transmission of warmth from warmer to cooler portions of a body ensuing in balancing of temperature. Owing to importance of thermal conductivity in engineering technologies, a finite difference scheme is developed to study the originality of thermal conductivity in a two-dimensional fluid motion in assembly with thermophoretic forces, variable thermal conductivity and viscous dissipative heat over a permeable horizontal surface. The thermophoretic effect is included in the concentration boundary layer equation and the formulation has adopted by Talbot-Cheng-Scheffer-Willis (1980). A suitable similarity transformation is adapted to convert the leading PDEs to non-linear ordinary differential equations in non-dimensional form. A well-tested, numerically stable finite difference scheme in connection with Bvp4c is employed via MATLAB code for the conservation of equations under the appropriate transformed boundary conditions. The impact of thermophoretic forces and thermal conductivity in presence of suction/blowing over the fluid velocity and temperature are significant. The thermal conductivity of a substantial is an important property that assistances in the growth of active boiler/refrigerating machineries. In this study, thermophoretic forces (TP) and thermal conductivity (β) enhances the fluid velocity in presence of blowing, while they declines the velocity due to suction. The validity and accuracy of the present model have been checked and found adequate agreement with the previous studies. The importance of such analysis over a horizontal surface have numerous manufacturing, industrial and engineering applications in plastic sheets extrusion, polymer extraction, blowing of glass, manufacture of paper, thermo-electronics and rubber sheets.

An energy stable finite difference scheme for the Ericksen–Leslie system with penalty function and its optimal rate convergence analysis

An energy stable finite difference scheme for the Ericksen–Leslie system with penalty function and its optimal rate convergence analysis

A well‐balanced and entropy stable scheme for a reduced blood flow model

AbstractA well‐known reduced model of the flow of blood in arteries can be formulated as a strictly hyperbolic system of two scalar balance laws in one space dimension where the unknowns are the cross‐sectional area of the artery and the average blood flow velocity as functions of the axial coordinate and time. This system is endowed with an entropy pair such that solutions of the balance equations satisfy an entropy inequality in the distributional sense. It is demonstrated that this property can be utilized to construct an entropy stable finite difference scheme for the blood flow model based on the general framework by Tadmor (E. Tadmor, Math. Comput., 49 (1987), 91–103). Furthermore, a fourth‐order extension of the resulting entropy conservative flux and a fourth‐order sign‐preserving reconstruction of the scaled entropy variables are employed as well as a second‐order strong stability preserving Runge–Kutta method for time discretization. The result is a computationally inexpensive and easy‐to‐implement explicit entropy stable scheme for the blood flow model. It is proven that the scheme is well‐balanced (i.e., preserves certain steady solutions of the model) and numerical examples are presented.

Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation

Hyperbolic partial differential equations are frequently referenced in modeling real-world problems in mathematics and engineering. Therefore, in this study, an initial-boundary value issue is proposed for the pseudo-hyperbolic telegraph equation. By operator method, converting the PDE to an ODE provides an exact answer to this problem. After that, the finite difference method is applied to construct first-order finite difference schemes to calculate approximate numerical solutions. The stability estimations of finite difference schemes are shown, as well as some numerical tests to check the correctness in comparison to the precise solution. The numerical solution is subjected to error analysis. As a result of the error analysis, the maximum norm errors tend to decrease as we increase the grid points. It can be drawn that the established scheme is accurate and effective

Stability of finite difference schemes for two-space dimensional telegraph equation

Stability of finite difference schemes for two-space dimensional telegraph equation

Weak solution to a Robin problem of anomalous diffusion equations: Uniqueness and stable algorithm for the TPC system

A Riemann‐Liouville fractional Robin boundary‐value problem is proposed to describe the fast heat transfer law both within isotropic materials and through the boundary of the materials in high temperature environment. The variational formulation of the fractional model is derived, and further, the energy estimation of the weak solution is deduced. Subsequently, the uniqueness theorem of weak solution is proved. A valid and stable finite difference scheme is developed for the fractional model. The numerical experiments are implemented to indicate that the fractional model is applicable to discover the thermal superdiffusion in the thermal protective clothing (TPC) system and numerical algorithms are effective to improve the intelligence of TPC design.

Numerical solutions of generalized Rosenau–KDV–RLW equation by using Haar wavelet collocation approach coupled with nonstandard finite difference scheme and quasilinearization

AbstractIn this article, we analyze and propose to compute the numerical solutions of a generalized Rosenau–KDV–RLW (Rosenau‐Korteweg De Vries‐Regularized Long Wave) equation based on the Haar wavelet (HW) collocation approach coupled with nonstandard finite difference (NSFD) scheme and quasilinearization. In the process of the numerical solution, the NSFD scheme is applied to discretize the first‐order time derivative, Haar wavelets are applied on spatial derivatives and the non‐linear term is taken care by quasilinearization technique. To discuss the efficiency of the method we compute error and error. We also use discrete mass and energy conservation to check the accuracy of the proposed methodology. The computed results have been compared with the existing methods, for example, three‐level average implicit finite difference technique, B‐spline collocation, three‐level linear conservative implicit finite difference scheme and conservative fourth‐order stable finite difference scheme.

Numerical analysis of energy-stable Crank-Nicolson finite difference schemes for the phase-field equation

Two stable finite difference schemes, with the second-order Crank-Nicolson/Adams-Bashforth method in time and the second-order finite difference approach in space, are proposed for solving the phase-field Allen-Cahn equation with a small parameter perturbation and strong nonlinearity. First, we design a finite difference scheme with conditional energy stability for the Allen-Cahn equation, which preserves the energy-decreasing property, and prove its error estimate. Second, by introducing an artificial stability term, we establish an unconditional energy-stable finite difference scheme for the Allen-Cahn equation, discuss its unconditional energy stability, and obtain its error estimate. Finally, numerical examples are presented to verify all of the theoretical results for the proposed schemes.

Strictly convex entropy and entropy stable schemes for reactive Euler equations

This paper presents entropy analysis and entropy stable (ES) finite difference schemes for the reactive Euler equations with chemical reactions. For such equations we point out that the thermodynamic entropy is no longer strictly convex. To address this issue, we propose a strictly convex entropy function by adding an extra term to the thermodynamic entropy. Thanks to the strict convexity of the proposed entropy, the Roe-type dissipation operator in terms of the entropy variables can be formulated. Furthermore, we construct two sets of second-order entropy preserving (EP) numerical fluxes for the reactive Euler equations. Based on the EP fluxes and the Roe-type dissipation operators, high-order EP/ES fluxes are derived. Numerical experiments validate the designed accuracy and good performance of our schemes for smooth and discontinuous initial value problems. The entropy decrease of ES schemes is verified as well.

Linear and fully decoupled scheme for a hydrodynamics coupled phase-field surfactant system based on a multiple auxiliary variables approach

We propose a linear, fully decoupled, and energy stable finite difference scheme for solving a phase-field surfactant fluid system. Inspired by the idea of multiple scalar auxiliary variables (MSAV) approach, two scalar auxiliary variables are used to transform the original governing equations into their equivalent forms. Based on the equivalent system, a highly efficient scheme can be developed. In one time cycle, the proposed algorithm can be efficiently performed, i.e., the surfactant ψ is explicitly updated, then the phase-field function ϕ, velocity field u, and pressure field p can be computed by solving linear systems with constant coefficients. The energy dissipation law for a modified energy can be estimated by using the proposed method. Various computational simulations confirm that the proposed method is not only accurate and energy stable but also works well for simulating surfactant-laden droplet dynamics.

On the stability estimates and numerical solution of fractional order telegraph integro-differential equation

We consider an initial-boundary value problem for fractional order telegraph integro-differential equation. In this study, fractional derivative can be considered in both Riemann-Liouville and Caputo senses since our initial condition is zero. We calculate approximate numerical solutions by constructing first and second order finite difference schemes. Estimations of the stability of finite difference scheme is given and also for some fractional orders, numerical examples are given to confirm the accuracy of the established difference scheme with respect to exact solution. Also some plots are presented for large values of x and t. Finally we show that our solutions continuously depend on the fractional derivatives by plotting error graph while increasing fractional order.

A positivity-preserving, energy stable scheme for a ternary Cahn-Hilliard system with the singular interfacial parameters

In this paper, we construct and analyze a uniquely solvable, positivity preserving and unconditionally energy stable finite-difference scheme for the periodic three-component Macromolecular Microsphere Composite (MMC) hydrogels system, a ternary Cahn-Hilliard system with a Flory-Huggins-deGennes free energy potential. The proposed scheme is based on a convex-concave decomposition of the given energy functional with two variables, and the centered difference method is adopted in space. We provide a theoretical justification that this numerical scheme has a pair of unique solutions, such that the positivity is always preserved for all the singular terms, i.e., not only two phase variables are always between 0 and 1, but also the sum of two phase variables is between 0 and 1, at a point-wise level. In addition, we use the local Newton approximation and multigrid method to solve this nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, energy dissipation and mass conservation properties.

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.

A new stable finite difference scheme and its error analysis for two‐dimensional singularly perturbed convection–diffusion equations

AbstractThis work focuses on the numerical solution of two‐dimensional singularly perturbed convection–diffusion equations via a new stable finite difference (NSFD) scheme on a tensor product of two piecewise‐uniform Shishkin meshes. First, we convert the two‐dimensional equation into two one‐dimensional equations using the alternating direction implicit technique. A NSFD scheme has been developed using Taylor's series and the one‐dimensional equations. Here the truncation of Taylor's series is different from the classical finite difference scheme. The convergence analysis is also studied on a tensor product of two piecewise‐uniform Shishkin meshes. Numerical simulations confirm the theory. Moreover, from the numerical illustrations, it is also observed that the method is parameter uniform convergent.

Convergence of energy stable finite-difference schemes with interfaces

We extend the convergence results in Svärd and Nordström (2019) [7] for single-domain energy-stable high-order finite difference schemes, to include domains split into several grid blocks. The analysis also demonstrates that reflective boundary conditions enjoy the same convergence properties. Finally, we briefly indicate that these results (and the previous ones in [7]) also hold in multiple dimensions.