Crank-Nicholson Scheme Research Articles

Development and Optimisation of a DNS Solver Using Open-source Library for High-performance Computing

We develope an optimised Direct Numerical Simulation (DNS) solver for laminar, transition, and turbulent flow. It utilises the Crank Nicholson scheme for time advancement, and it is spatially accurate up to fourth-order on uniform grids. The solver's salient features are multi-dimensional parallelisation and modular-based code, which is written in object-oriented C++ language. The DNS solver is developed in the MPI parallel platform using a cubic decomposition technique. The open-source library used for optimisation is MPI, blitz++, YAML, and HDF5 library. The parallel performance analysis of the optimised DNS solver on the multiprocessor computing system (supercomputer) shows 80 efficiency. The solver is verified and benchmarked for wall-bounded flows by simulation of laminar, transition, and turbulent flow in a square duct.

Chemically reacting viscous fluid flow on a permeable cylinder susceptible to oscillations

An analysis is accomplished for a natural convective flow of an isochoric and chemically reactive viscid fluid around an impetuously commenced oscillating vertical cylinder encapsulated in a porous structure. The governing problem comprising of a system of conjoined nonlinear boundary layer equations with adapted boundary conditions are solved numerically by a Crank-Nicholson scheme. This scheme is one among the finite difference schemes which is an implicit, unconditionally stable and convergent scheme. The impacts of appurtenant parameters of an oscillating cylinder on velocity, temperature, concentration, viscous drag, exchange of thermal energy and mass transfer are presented graphically and interpreted. Major findings of the study disclose the fact that oscillations with higher phase angle reinforce the rate of heat and mass transfer. The behaviors of the streamlines and isotherms of the fluid are also presented and discussed.

Perfectly matched layers for the Dirac equation in general electromagnetic texture.

Perfectly matched layer (PML) boundary conditions are constructed for the Dirac equation and general electromagnetic potentials. A PML extension is performed for the partial differential equation and two versions of a staggered-grid single-cone finite-difference scheme. For the latter, PML auxiliary functions are computed either within a Crank-Nicholson scheme or one derived from the formal continuum solution in integral form. Stability conditions are found to be more stringent than for the original scheme. Spectral properties under spatially uniform PML confirm damping of any out-propagating wave contributions. Numerical tests deal with static and time-dependent electromagnetic textures in the boundary regions for parameters characteristic for topological insulator surfaces. When compared to the alternative imaginary-potential method, PML offers vastly improved wave absorption owing to a more efficient suppression of back-reflection. Remarkably, this holds for time-dependent textures as well, making PML a useful approach for transient transport simulations of Dirac fermion systems.

Crank-Nicholson difference scheme for the system of nonlinear parabolic equations observing epidemic models with general nonlinear incidence rate.

In this work, we study second order Crank-Nicholson difference scheme (DS) for the approximate solution of problem (1). The existence and uniqueness of the theorem on a bounded solution of Crank-Nicholson DS uniformly with respect to time step τ is proved. In practice, theoretical results are presented on four systems of nonlinear parabolic equations to explain how it works on one and multidimensional problems. Numerical results are provided.

Higher order Galerkin–collocation time discretization with Nitsche’s method for the Navier–Stokes equations

We propose and study numerically the implicit approximation in time of the Navier–Stokes equations by a Galerkin–collocation method in time combined with inf–sup stable finite element methods in space. The conceptual basis of the Galerkin–collocation approach is the establishment of a direct connection between the Galerkin method and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs in terms of less complex algebraic systems of the latter. Regularity of higher order in time of the discrete solution is ensured further. As an additional ingredient, we employ Nitsche’s method to impose all boundary conditions in weak form with the perspective that evolving domains become feasible in the future. We carefully compare the performance properties of the Galerkin–collocation approach with a standard continuous Galerkin–Petrov method using piecewise linear polynomials in time, that is algebraically equivalent to the popular Crank–Nicholson scheme. The condition number of the arising linear systems after Newton linearization as well as the reliable approximation of the drag and lift coefficient for laminar flow around a cylinder (DFG flow benchmark with Re=100; cf. (Turek and Schäfer, 1996)) are investigated. The superiority of the Galerkin–collocation approach over the linear in time, continuous Galerkin–Petrov method is demonstrated therein.

Numerical Scheme for Solving Time–Space Vibration String Equation of Fractional Derivative

In this paper, we present a numerical scheme and alternating direction implicit scheme for the one-dimensional time–space fractional vibration equation. Firstly, the considered time–space fractional vibration equation is equivalently transformed into their partial integro-differential forms by using the integral operator. Secondly, we use the Crank–Nicholson scheme based on the weighted and shifted Grünwald–difference formula to discretize the Riemann–Liouville and Caputo derivative, also use the midpoint formula to discretize the first order derivative. Meanwhile, the classical central difference formula is applied to approximate the second order derivative. The convergence and unconditional stability of the suggested scheme are obtained. Finally, we present an example to illustrate the method.

Mathematical estimation of fluid concentration in human skin during water immersion

IntroductionThe concentration of fluid and its analysis in human skin is innately a challenge due to its continuous movement and involvement in maximum life processes. The concentration of the fluid gets affected by the diffusion of fluids through the skin, which acts as the main barrier between the human body and the external environment. Therefore, it becomes imperative to study the process and impact of the diffusion of fluids through the skin. The problem becomes more interesting when the human body is immersed in water. ObjectivesThe present paper studies the change in the fluid distribution of human skin during its immersion in water of different temperatures. The application part of the paper visualizes various impaired vascular function and muscle soreness by water immersion during the physiotherapy treatment. MethodsA mathematical model based on the two-dimensional diffusion equation, along with appropriate boundary conditions, has been formulated. The maximum of the relevant parameters, such as fluid regulation, transfer coefficient, evaporation rate, etc., influencing the fluid distribution, have been incorporated. The model has been solved by variational finite element method, and numerical results have been obtained by the Crank-Nicholson scheme. ResultsThe increase in fluid concentration due to treatment with cold and acute hot water immersion has been noted, and the role of water immersion in enhancing the recovery in exercise-induced muscular damage has been analyzed. ConclusionsThe paper addressed the issue of rate of water diffusion through human skin, which otherwise couldn't be drawn from the analogy of gas diffusion through the membrane due to the variation in permeabilities of the two processes. The paper has applications in water immersion therapies and other activities like monitoring swimming induced pulmonary edema, etc.

On the stability analysis of the time-fractional variable order Klein-Gordon equation and a numerical simulation

In this paper, the Klein - Gordon equation is generalized using the concept of the variational order derivative. We try to construct the Crank-Nicholson scheme for numerical solutions of the modified Klein- Gordon equation. Stability analysis of the Crank-Nicholson scheme is examined and analyzed to prove the proposed method is stable for solving the time-fractional variable order Klein- Gordon equation. A numerical example is also given for illustration.

Higher Order Compact Schemes for the Spatial Discretization of Linear Parabolic Partial Differential Equations

A system of compact schemes used, to approximate the partial derivative of Linear Parabolic Partial Differential Equations (LPPDE), on the non-boundary nodes. Euler time integration for the temporal derivative, the Crank-Nicholson (CN)scheme for the spatial derivatives and the source term are used. The higher order spatial accuracy of the developed, central difference based compact schemes for the one and two-dimensional diffusion equations are validated, by solving numerically two test problems. The compact scheme based calculations involve a tridiagonal matrix vector multiplication and a vector vector subtraction. The interest is to demonstrate the higher order spatial accuracy and the better rate of convergence of the solution produced using the developed compact 4th order scheme, when compared with the same produced, using the conventional 2nd order scheme. The assessment is made, in terms of the discrete or norm of the true error of the converged numerical solution.

Sediment transport and bed evolution model for complex river systems.

This work presents the theoretical background, development, and preliminary evaluation of a one-dimensional unsteady sediment transport and bed evolution model for a looped river network. The sediment transport and bed evolution model employs a concept that differentiates sediment particles moving in the form of suspended sediment, and near bed, and bed sediment. Applying the active layer concept, the developed model utilizes the appropriate exchange mechanisms between the suspended sediment and active layer material, which are presented in great detail. The governing transport equations were solved using the split operator approach that resulted in two successive steps. The advection equations were solved using the characteristics method, whereas the diffusion equations were discretized using the Crank-Nicholson scheme. The obtained system was complemented with additional auxiliary equations in order to allow sediment transport and bed evolution simulation at nodes, thus enabling the same in a looped river network. The derived equations were applied to develop an open-channel flow, sediment transport, and bed evolution model that was subjected to a series of preliminary numerical tests.

Numerical simulation of water hammer using implicit Crank-Nicolson Local Multiquadric Based Differential Quadrature

This paper focuses on developing a numerical model based on Implicit Local Multiquadric Based Differential Quadrature Method for simulation of water hammer phenomena in a pipeline system consisting of a valve, pipe and surge tank. This model is used with the aim of reducing computational costs and increasing accuracy. After brief interdicting of governing equations of water hammer, the spatial and temporal derivatives are implicitly discretized by the Local Multiquadric Based Differential Quadrature and Crank Nicholson scheme, respectively. The system of governing equations of the water hammer is discretized in matrix form and solved by applying appropriate boundary and initial conditions. The proposed model is verified by two experimental case. The comparison between obtained results and experimental reported data indicates that the presented method is in good agreement with experimental observations. To evaluate the computational cost, these examples were also solved concurrently with Method of Characteristics. The comparison showed that this method is able to simulate the water hammer more accurately despite less computational effort. It is also found the accuracy of the model depends on the Courant number. However, the model maintains its stability in Courant numbers more than the one.

Waves in Two Coaxial Elastic Cubically Nonlinear Shells with Structural Damping and Viscous Fluid Between Them

This article investigates longitudinal deformation waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid between them. The presence of a viscous incompressible fluid between the shells, as well as the influence of the inertia of the fluid motion on the amplitude and velocity of the wave, are taken into account. The mathematical model phenomenon is constructed by means of the method of two-scale asymptotic expansion. Structural damping in the shells and surrounding elastic media did not allow discovery of the exact solution of the problem of the deformation waves propagation. This leads to the need for numerical methods. A numerical study of the model constructed in the course of this work is carried out by using a difference scheme for the equation similar to the Crank–Nicholson scheme for the heat equation. In the absence of the structural damping and surrounding media influences, and under the similar initial conditions for both shells, the velocity and amplitude of the wave do not change. The result of the numerical experiment coincides with the exact solution, which is found in the case of the absence of the structural damping and surrounding media influences; therefore, the difference scheme is adequate to the generalized modified Korteweg–de Vries equations system. There is energy is transferred in the presence of the fluid, between the shells. The presence of inertia of the fluid motion leads to a decrease in the velocity of the deformation wave.

On Solutions of Fractional order Telegraph Partial Differential Equation by Crank-Nicholson Finite Difference Method

Abstract The exact solution is calculated for fractional telegraph partial differential equation depend on initial boundary value problem. Stability estimates are obtained for this equation. Crank-Nicholson difference schemes are constructed for this problem. The stability of difference schemes for this problem is presented. This technique has been applied to deal with fractional telegraph differential equation defined by Caputo fractional derivative for fractional orders α = 1.1, 1.5, 1.9. Numerical results confirm the accuracy and effectiveness of the technique.

A discontinuous Galerkin method for a two dimensional reduced resistive MHD model

We are concerned with the numerical approximation of an incompressible ionized gas (plasma) flowing in a toroidal geometry. We also assume that the flow is independent of the toroidal coordinate and the resulting model is thus 2-D. We consider a symmetric formulation, the so-called Reduced Resistive MHD model, where the governing equation gives the evolution of the axial current and vorticity (scalar variables). These equations are written in a quasi conservative form and, using the Discontinuous Galerkin (DG) framework, an approximation strategy is proposed and analyzed for triangular meshes. This approach combines a Galerkin projection of the velocity stream function and magnetic flux to obtain divergence-free approximations, together with a DG approximation of the evolutionary equations for current and vorticity, while the integration is performed using Crank–Nicholson scheme. The designed methodology is validated on the kink-mode and the tilting MHD instabilities.

Condition Monitoring a Coupled Differential Equation Using Order of Accuracy of Boundary Values in a Lid-Driven Cavity

The unsteady vorticity-transport equation and coupled stream function equation are discretised using Crank–Nicholson scheme in a finite difference mesh. The boundary conditions following Thom’s formula, Jenison’s formula of second and third order and computational boundary method are applied to solve the coupled equations. The geometry for the problem is a lid-driven cavity in which the coupled differential equations are solved. The order of accuracy of the boundary conditions affects largely the value of vorticity values at the centre of the primary vortex. At lower Reynolds number these values don’t have any impact; however at large Reynolds numbers those values are affected by large amount. Sometimes the computed values overshot the theoretical value of vorticity i.e. − 1.88596 with increase of Reynolds number. For this computation the grid meshes 129 × 129 and 257 × 257 are used. It is also observed that the computed vorticity value remain within the theoretical limit for the Jenison’s second order, third order and computational boundary element method.

A compact Crank–Nicholson scheme for the numerical solution of fuzzy time fractional diffusion equations

Fuzzy fractional partial differential equations are a generalization of classical fuzzy partial differential equation which can, in certain circumstances, provide a better explanation for certain phenomena. In this paper, a compact Crank–Nicholson scheme is developed, analyzed and applied to solve the fuzzy time diffusion equation with fractional order $$0 < \alpha \le 1$$ . It is based on double parametric form of fuzzy numbers and the time fractional derivative is defined using the Caputo definition. The truncation error and stability of the proposed scheme is discussed. The compact Crank–Nicholson scheme is shown to be a feasible scheme via a numerical example.

Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control

Traveling waves remain significant in Applied Sciences mostly because they involve the movement of energy carrier particles. In this paper, traveling waves described by a generalized system, the fractional variable order hyperbolic Liouville model is solved numerically by means of Crank-Nicholson scheme. Detailed analysis are performed and prove that the numerical method is stable and converges. Simulations reveal that the model's variable order derivative (a function of time and position variables) has a considerable impact on the dynamics of the whole system. It influences the movement and the shape of the resulting waves including their amplitude, their wavelength as well as their compression and rarefaction processes. Such a variable order derivative becomes, due to these results, a substantial parameter and non-constant tool for the regulation and control of models describing wave motion.

Numerical investigation of flow and combined natural-forced convection from an isothermal square cylinder in cross flow

Unsteady flow and heat transfer from a horizontal isothermal square cylinder is studied numerically using a three-dimensional computational model to investigate the influence of buoyancy on the forced flow and heat transfer characteristics. The numerical model is based on a horizontal square cylinder subjected to laminar fluid flow in an unconfined channel. The governing equations in 3D form are solved using a fractional step method based on the finite difference discretization in addition to a Crank–Nicholson scheme employed to the convective and the viscous terms. Two working fluids–air (Pr = 0.7) and water (Pr = 7)–are considered, and the flow and heat transfer simulations were carried out for the Reynolds and Richardson numbers in the intervals 55 ≤ Re ≤ 250 and 0 ≤ Ri ≤ 2, respectively. The flow characteristics such as time-averaged drag/lift, rms drag/rms lift coefficients as well as Strouhal number were computed. The heat transfer from the cylinder is assessed by mean Nusselt number (and rms Nusselt number) over the total heated cylinder walls. As the buoyancy increases, the mass and the velocity of the fluid flowing underneath the cylinder increases. The fluid is injected into the near wake region with an upward motion which significantly alters the flow field in the downstream as well as upstream regions. The effects of Reynolds, Richardson and Prandtl numbers on the flow field and temperature distributions are discussed in detail. It is shown that the flow and heat transfer characteristics are influenced more for air than water. To fill the void in the literature, useful empirical correlations of practical importance are derived for pure forced and pure natural as well as mixed convection. The mixed convection correlations, in terms of the ratio of pure forced convection, are also developed, and their implications are discussed.

Time-fractional variable-order telegraph equation involving operators with Mittag-Leffler kernel

ABSTRACTIn this paper, we have generalized the time-fractional telegraph equation involving operators with Mittag-Leffler kernel of variable order in Liouville-Caputo sense. The fractional variable-order equation was solved numerically via Crank-Nicholson scheme. We present the existence and uniqueness of the solution. Numerical simulations of the special solutions were done and new behaviors are obtained.

Time-accurate calculation and bifurcation analysis of the incompressible flow over a square cavity using variational multiscale modeling

A thorough variational multiscale (VMS) modeling of the Navier–Stokes equations is used to compute numerical solutions of the incompressible flow over an open cavity. This case features several competing instabilities, and is highly challenging for VMS methods with regard to frequency and pattern selection, because of the non-normality of the linearized Navier–Stokes operator. The relevance of the approach is thus carefully assessed by comparing to direct numerical simulation (DNS) data benchmarked at several Reynolds numbers, and highly accurate time advancing methods are shown to predict relevant evolutions of the transient and saturated solutions. The VMS reduces substantially the computational cost, by ∼35% (resp. ∼60%) in terms of CPU time using a semi-implicit discretization scheme based on backward differentiation formula (resp. the implicit Crank–Nicholson scheme), and by ∼80% in terms of memory requirement. Eventually, the highly efficient semi-implicit VMS numerical framework is used to unravel the onset of the flow oscillations and the selection of the limit cycle frequency, that happens to involve a subcritical Neimark–Sacker bifurcation.