Hybrid Finite Difference Scheme Research Articles

A second order numerical method for singularly perturbed Volterra integro-differential equations with delay

Abstract This study deals with singularly perturbed Volterra integro-differential equations with delay. Based on the properties of the exact solution, a hybrid difference scheme with appropriate quadrature rules on a Shishkin-type mesh is constructed. By using the truncation error estimate techniques and a discrete analogue of Grönwall’s inequality it is proved that the hybrid finite difference scheme is almost second order accurate in the discrete maximum norm. Numerical experiments support these theoretical results and indicate that the estimates are sharp.

Hybrid conservative central/WENO finite difference scheme for two-dimensional detonation problems

In the present work, a hybrid finite difference scheme is proposed and tested for the study of two-dimensional detonation problems. The hybrid scheme consists of a fifth-order weighted essentially nonoscillatory (WENO) method for discontinuous regions and a fourth or sixth-order robust conservative central finite difference scheme for the remaining domain. An arbitrary shock sensor is used to identify the discontinuities. The proposed shock sensor is based on the absolute value of the density gradient with an arbitrary threshold computed at each time step using a fast image segmentation technique. The sensor is complemented with a Ducros sensor to avoid the selection of vortices as discontinuities. The hybrid scheme is tested for several benchmark examples, including a two-dimensional transverse detonation wave, detonation wave diffraction, and a multiple obstacles test case. The obtained results show that the proposed sensor detects discontinuities adequately and the hybrid schemes give the same results as the fifth-order WENO in faster computational time.

Uniformly convergent hybrid numerical scheme for singularly perturbed turning point problems with delay

In the present work, we consider a class of singularly perturbed turning point problems with delay for their numerical solution. Due to the presence of a delay, there occurs an interior layer in the solution along with twin boundary layers (SIAM J. Appl. Math. 42(3): 502–530). Some a priori estimates are given on the exact solution and its derivatives. A numerical technique composed of hybrid finite difference scheme on a layer-adapted Shishkin mesh is proposed. We establish the consistency, stability and convergence of the proposed technique. To validate the theoretical predictions, we present some numerical illustrations. The proposed scheme is proved to have an almost second-order convergence rate, uniform with respect to the perturbation parameter ε.

Direct numerical simulation of compressible turbulence accelerated by graphics processing unit: An open-source high accuracy accelerated computational fluid dynamic software

This paper introduces open-source computational fluid dynamics software named open computational fluid dynamic code for scientific computation with graphics processing unit (GPU) system (OpenCFD-SCU), developed by the authors for direct numerical simulation of compressible wall-bounded turbulence. This software is based on the finite difference method and is accelerated by the use of a GPU, which provides an acceleration by a factor of more than 200 compared with central processing unit (CPU) software based on the same algorithm and a number of Message Passing Interface processes, and the running speed of OpenCFD-SCU with just 512 GPUs exceeds that of CPU software with 130 000 CPUs. GPU-Stream technology is used to implement overlap of computing and communication, achieving 98.7% parallel weak scalability with 24 576 GPUs. The software includes a variety of high-precision finite difference schemes and supports a hybrid finite difference scheme, enabling it to provide both robustness and high precision when simulating complex supersonic and hypersonic flows. When used with the wide range of supercomputers currently available, the software should be able to improve the performance of large-scale simulations by up to two orders on the computational scale. Then, OpenCFD-SCU is applied to a validation and verification case of a Mach 2.9 compression ramp with mesh numbers up to 31.2 × 109.

Analysis of a Hybrid Numerical Scheme for Singularly Perturbed Convection-Diffusion Type Delay Problems

In this paper, a class of singularly perturbed delay differential equations of convection-diffusion (C–D) type is considered. The solutions for these problems exhibit boundary as well as interior layer due to the presence of delay. A few a priori estimates are given on the exact solution and its derivatives. We propose a numerical technique consisting of hybrid finite difference scheme on a Shishkin-type mesh. The proposed technique is analyzed for its consistency, stability and convergence. To show the effectiveness of the present approach, we conduct some numerical experiments. An almost second-order [Formula: see text]-uniform convergence is established.

A Hybrid Finite Difference Method for Singularly Perturbed Delay Partial Differential Equations with Discontinuous Coefficient and Source

The article presents a hybrid finite difference scheme to solve a singularly perturbed parabolic functional differential equation with discontinuous coefficient and source. The simultaneous presence of deviating argument with a discontinuous source and coefficient makes the problem stiff. The solution of the problem exhibits turning point behaviour across discontinuity as tends to zero. The hybrid scheme presented is a composition of a central difference scheme and a midpoint upwind scheme on a specially generated mesh. At the same time, an implicit finite difference method is used to discretize the time variable. Consistency, stability, and convergence of the presented numerical approach have been investigated. The presented method converges uniformly independent of the perturbation parameter. Numerical results have been presented for two test examples that verify the effectiveness of the scheme.

Efficient High-Order Discretization Methods for Coupled Systems of Singularly Perturbed Parabolic Convection-Diffusion Problems

This paper develops efficient high-order convergent discretization methods for coupled systems of singularly perturbed parabolic convection-diffusion problems exhibiting boundary layers. The discretization of the problem consists of two splitting schemes in time on a uniform mesh and a high-order hybrid finite difference scheme in space on appropriate layer-adapted Shishkin mesh. The splitting schemes decoupled the vector approximate solution at each time level, resulting in a reduced computational time. The convergence of the discretization methods is discussed in two steps using the barrier function approach. The proposed methods are proved to be uniformly convergent, having first-order accuracy in time and almost second-order accuracy in space. The splitting schemes are very efficient in comparison with the standard discretization. Numerical results confirm the theoretical findings and illustrate the efficiency of the proposed discretization methods.

Hybrid Finite Difference Schemes for Elliptic Interface Problems with Discontinuous and High-Contrast Variable Coefficients

For elliptic interface problems with discontinuous coefficients, the maximum accuracy order for compact 9-point finite difference scheme in irregular points is three [7]. The discontinuous coefficients usually have abrupt jumps across the interface curve in the porous medium of realistic problems, causing the pollution effect of numerical methods. So, to obtain a reasonable numerical solution of the above problem, the higher order scheme and its effective implementation are necessary. In this paper, we propose an efficient and flexible way to achieve the implementation of a hybrid (9-point scheme with sixth order accuracy for interior regular points and 13-point scheme with fifth order accuracy for interior irregular points) finite difference scheme in uniform meshes for the elliptic interface problems with discontinuous and high-contrast piecewise smooth coefficients in a rectangle $\Omega$. We also derive the $6$-point and $4$-point finite difference schemes in uniform meshes with sixth order accuracy for the side points and corner points of various mixed boundary conditions (Dirichlet, Neumann and Robin) of elliptic equations in a rectangle. Our numerical experiments confirm the flexibility and the sixth order accuracy in $l_2$ and $l_{\infty}$ norms of the proposed hybrid scheme.

An almost second order hybrid scheme for the numerical solution of singularly perturbed parabolic turning point problem with interior layer

In this article, we consider a one dimensional singularly perturbed parabolic convection–diffusion problem with interior turning point. The convection coefficient of the considered problem is vanishing inside the spatial domain and also, exhibits an interior layer. As a result, the exact solution of the considered problem contains an interior layer. A higher order numerical method is constructed and analyzed for the numerical solution of the considered problem. To discretize the time direction, we have used the classical implicit Euler method on a uniform mesh. Also, a hybrid finite difference scheme is employed on a generalized Shishkin mesh condensing in the interior layer region to discretize the spatial domain. Rigorous analysis is performed to show that the proposed method is ε-uniformly convergent of order almost two. The higher accuracy and convergence rate of the proposed scheme are verified via implementing numerical experiments on two test problems. Comparison is done with the scheme proposed in O’Riordan and Quinn (2015) for the considered class of problems.

Spline approximation method for singularly perturbed differential-difference equation on nonuniform grids

In this paper, a second-order singularly perturbed differential-difference equation involving mixed shifts is considered. At first, through Taylor series approximation, the original model is reduced to an equivalent singularly perturbed differential equation. Then, the model is treated by using the hybrid finite difference scheme on different types of layer adapted meshes like Shishkin mesh, Bakhvalov–Shishkin mesh and Vulanović mesh. Here, the hybrid scheme consists of a cubic spline approximation in the fine mesh region and a midpoint upwind scheme in the coarse mesh region. The error analysis is carried out and it is shown that the proposed scheme is of second-order convergence irrespective of the perturbation parameter. To display the efficacy and accuracy of the proposed scheme, some numerical experiments are presented which support the theoretical results.

An Efficient Low-Dissipation Hybrid Central/WENO Scheme for Compressible Flows

This paper develops a new hybrid finite difference scheme which consists of a nonlinear WENOCU4 scheme to maintain the numerical stability near discontinuities and a fourth-order linear central scheme elsewhere to accurately resolve smooth fluctuations and to speed up the computation. The central scheme is constructed in a robust skew-symmetric form which satisfies energy conservation property. A new efficient discontinuity detector is employed to identify the smoothness of the flowfield. As for the one-dimensional scalar equations, the hybrid scheme behaves almost no attenuation associated with propagation error. The quantitative analysis of the shock-capturing error is investigated. The hybrid scheme also exhibits high-resolution spectral property in the wavenumber space. Extensive cases of Euler equations further demonstrate the high-resolution shock-capturing ability of the detector, remarkable vortices-resolving capacity, and superior computational efficiency of the hybrid scheme. The reduced computational cost of the hybrid scheme is about 30−50% with respect to the baseline scheme.

Uniformly Convergent New Hybrid Numerical Method for Singularly Perturbed Parabolic Problems with Interior Layers

In this article, we deal with a class of singularly perturbed parabolic convection–diffusion initial-boundary-value problems having discontinuous convection coefficient. Aiming to get better numerical approximation to the solutions of this class of problems, we devise a new hybrid finite difference scheme on a layer-resolving piecewise-uniform Shishkin mesh in the spatial direction, and the time derivative is discretized by the backward-Euler method in the temporal direction. We discuss the stability of the proposed method and establish the parameter-uniform error estimate. Numerical results are also displayed to support the theoretical findings and compared with the existing hybrid scheme to show the improvement in terms of spatial order of convergence. Further, we carry out numerical experiment for the semi-linear parabolic initial-boundary-value problems.

Wave Fields and Nearshore Currents in the Coastal Region Opposite San Mauro Cilento (Italy)

In this paper in order to simulate nearshore currents in computational domains representing the complex morphology of real coastal regions we use a model based on a contravariant integral form of the fully nonlinear Boussinesq equations (FNBE). The contravariant integral form, in which Christoffel symbols are absent, of the continuity equation does not contain the dispersive term. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations (NSWE). The capacity of the proposed model to correctly simulate the wave train propagation on a highly distorted grid is verified against test case present in the literature. The simulation of wave fields and nearshore currents in the coastal region, opposite San Mauro Cilento (Italy) in presence of a system of T-head groins, is numerically reproduced by using the proposed model.

Uniform Hybrid Difference Scheme for Singularly Perturbed Differential-Difference Turning Point Problems Exhibiting Boundary Layers

In this paper, a class of linear second-order singularly perturbed differential-difference turning point problems with mixed shifts exhibiting two exponential boundary layers is considered. For the numerical treatment of these problems, first we employ a second-order Taylor’s series approximation on the terms containing shift parameters and obtain a modified singularly perturbed problem which approximates the original problem. Then a hybrid finite difference scheme on an appropriate piecewise-uniform Shishkin mesh is constructed to discretize the modified problem. Further, we proved that the method is almost second-order ɛ-uniformly convergent in the maximum norm. Numerical experiments are considered to illustrate the theoretical results. In addition, the effect of the shift parameters on the layer behavior of the solution is also examined.

A higher-order hybrid numerical scheme for singularly perturbed convection-diffusion problem with boundary and weak interior layers

In this paper, we study the numerical solutions of singularly perturbed convection-diffusion two-point BVP as well as one-dimensional parabolic convection-diffusion IBVP with discontinuous convection coefficient (positive throughout the domain) and source term. The analytical solutions of these kind of problems exhibit a boundary layer near x = 0 and a weak interior layer near x = ξ. We discretise the spatial domain by the piecewise-uniform Shishkin mesh and the temporal domain by a uniform mesh. To approximate the spatial derivatives, we apply the hybrid finite difference scheme. The implicit-Euler scheme is used for discretising the temporal derivative. For the time independent problem, we derive that the proposed hybrid scheme is e-uniformly convergent of almost second-order and for the time dependent problem, we also prove that the proposed scheme is e-uniformly convergent of almost second-order in space and first-order in time. To validate the theoretical estimates, some numerical results are presented.

A higher-order hybrid numerical scheme for singularly perturbed convection-diffusion problem with boundary and weak interior layers

In this paper, we study the numerical solutions of singularly perturbed convection-diffusion two-point BVP as well as one-dimensional parabolic convection-diffusion IBVP with discontinuous convection coefficient (positive throughout the domain) and source term. The analytical solutions of these kind of problems exhibit a boundary layer near x = 0 and a weak interior layer near x = ξ. We discretise the spatial domain by the piecewise-uniform Shishkin mesh and the temporal domain by a uniform mesh. To approximate the spatial derivatives, we apply the hybrid finite difference scheme. The implicit-Euler scheme is used for discretising the temporal derivative. For the time independent problem, we derive that the proposed hybrid scheme is ε-uniformly convergent of almost second-order and for the time dependent problem, we also prove that the proposed scheme is ε-uniformly convergent of almost second-order in space and first-order in time. To validate the theoretical estimates, some numerical results are presented.

A parameter uniform numerical scheme for singularly perturbed differential-difference equations with mixed shifts

In this paper, we consider a second-order singularly perturbed differential-difference equations with mixed delay and advance parameters. At first, we approximate the model problem by an upwind finite difference scheme on a Shishkin mesh. We know that the upwind scheme is stable and its solution is oscillation free, but it gives lower order of accuracy. So, to increase the convergence, we propose a hybrid finite difference scheme, in which we use the cubic spline difference method in the fine mesh regions and a midpoint upwind scheme in the coarse mesh regions. We establish a theoretical parameter uniform bound in the discrete maximum norm. To check the efficiency of the proposed methods, we consider test problems with delay, advance and the mixed parameters and the results are in agreement with our theoretical findings.

Evolution of the Diffusion-Induced Flow over a Disk Submerged in a Stratified Viscous Fluid

The paper presents the results of the mathematically simulated evolution of a 3D diffusion-induced flow over a disk (with diameter d and thickness H = 0.76 d) immersed in a linearly density-stratified incompressible viscous fluid (described by a system of the Navier–Stokes equations in the Boussinesq approximation). The disk rests at the level of the neutral buoyancy (coinciding with its symmetry axis z) and disturbs the homogeneity of the background diffusion flux in the fluid forming a complex system of slow currents (internal gravitational waves). Over time, two thin horizontal convection cells are formed at the upper and lower parts of the disk stretching parallel to the z axis and adjacent to the base cell with thickness d/2. This work is the first to analyze in detail the fundamental mechanism for the formation of each new half-wave near the vertical axis x (passing through the center of the disk) during half the buoyancy period of the fluid Tb. This mechanism is based on gravitational instability. The emergence of this instability is first detected at 0.473Tb at a height of 3.9 d above the center of the disk. The same mechanism is also implemented over the place where the body moves in the horizontal direction. The 3D vortex structure of the flow is visualized by constructing the isosurfaces of the imaginary part of the complex-conjugate eigenvalues of the velocity gradient tensor. In mathematical simulation, we employed a numerical method SMIF that has proved itself over three decades with an explicit hybrid finite-difference scheme for the approximation of the convective terms of the equations (second-order approximation, monotonicity).

A parameter-uniform hybrid finite difference scheme for singularly perturbed system of parabolic convection-diffusion problems

ABSTRACT We present the hybrid finite difference scheme for singularly perturbed system of parabolic convection-diffusion problems exhibiting overlapping boundary layers. We discretize the time derivative by the backward-Euler method and the spatial derivatives is discretized by the hybrid difference scheme on Shishkin mesh. We have shown that the presented numerical scheme is parameter-uniform convergent of first-order in temporal variable and almost second-order in spatial variable. Numerical experiments supporting the theoretical results are presented.

Parameter-uniform fractional step hybrid numerical scheme for 2D singularly perturbed parabolic convection–diffusion problems

This article focuses on developing and analyzing an efficient numerical scheme for solving two-dimensional singularly perturbed parabolic convection–diffusion initial-boundary value problems exhibiting a regular boundary layer. For approximating the time derivative, we use the Peaceman–Rachford alternating direction implicit method on uniform mesh and for the spatial discretization, a hybrid finite difference scheme is proposed on a special rectangular mesh which is tensor-product of piecewise-uniform Shishkin meshes in the spatial directions. We prove that the numerical scheme converges uniformly with respect to the perturbation parameter $$\varepsilon $$ and also attains almost second-order spatial accuracy in the discrete supremum norm. Finally, numerical results are presented to validate the theoretical results. In addition to this, numerical experiments are conducted to demonstrate the effect of the time-dependent boundary conditions in the order of convergence numerically by introducing the classical evaluation of the boundary data; and also the improvement in the spatial order of accuracy of the present method by considering the Bakhvalov–Shishkin mesh in the spatial directions.