Search Results (13)

In this paper, high-order compact difference methods (HOCDMs) are proposed to solve the semi-linear Sobolev equations (SLSEs), which arise in various physical models, such as porous media flow and heat conduction. First, a two-level numerical method is given by applying the Crank–Nicolson (C-N) method in time and the fourth-order compact difference method in space. This method is shown to achieve second-order accuracy in time and fourth-order accuracy in space. Subsequently, we introduce the Richardson extrapolation technique to improve the temporal accuracy of the two-level method from second order to fourth order. Furthermore, we devise a fully fourth-order method in both time and space by applying the fourth-order difference method to discretize both temporal and spatial derivatives, and we provide a proof of its convergence. Finally, a series of numerical experiments is conducted to verify the effectiveness of the proposed methods. Full article

Two numerical methods are investigated for the initial–boundary value problem of a nonlinear Rosenau–KdV–RLW equation with homogeneous boundary conditions. With the premise of achieving second-order theoretical accuracy in the temporal direction, two-level linearization discretization and three-level extrapolated linearization discretization are applied to nonlinear terms, respectively. To achieve a higher theoretical accuracy in the spatial direction, the Richardson extrapolation combination technique is employed; thereby, a two-level linearized difference scheme and a three-level linear difference scheme for the Rosenau–KdV–RLW equation are proposed, both with a theoretical accuracy of $O({\tau }^2+h^4)$. The two-level difference scheme also reasonably simulates the conservation property of the problem. The convergence and stability of the two schemes are proven using mathematical induction and discrete functional analysis methods. The numerical results demonstrate the effectiveness of both schemes. Full article

In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and a past solution. The RE is a very simple and powerful technique that can be used to increase the order of accuracy of any approximation process by eliminating the lowest order error term(s) from its asymptotic error expansion. The spatial approximation of the governing Navier–Stokes equations is performed with a high-order accurate discontinuous Galerkin (dG) method. The presented numerical results for canonical test cases, i.e., the isentropic convecting vortex and the unsteady vortex shedding behind a circular cylinder, aim to assess the performance of the BE-BDF2 scheme, in its standard version and equipped with RE, by comparing it with the ones obtained by using more classical methods, like the BDF2, the second-order accurate Crank–Nicolson (CN2) and the explicit third-order accurate Strong Stability Preserving Runge–Kutta scheme (SSP-RK3). Full article

In several important scientific fields, the efficient numerical solution of symmetric systems of ordinary differential equations, which are usually characterized by oscillation and periodicity, has become an open problem of interest. In this paper, we construct a class of embedded exponentially fitted Rosenbrock methods with variable coefficients and adaptive step size, which can achieve third order convergence. This kind of method is developed by performing the exponentially fitted technique for the two-stage Rosenbrock methods, and combining the embedded methods to estimate the frequency. By using Richardson extrapolation, we determine the step size control strategy to make the step size adaptive. Numerical experiments are given to verify the validity and efficiency of our methods. Full article

This article highlights the inter-comparisons of the wind measurement techniques available in deep water areas working towards combining them to obtain optimal estimates of the wind power potential. More specifically, this article presents comparisons of the Ferry Lidar Experiment wind data with those of the Advanced Scatterometer (ASCAT), the FINO2 meteorological mast, and the New European Wind Atlas (NEWA) simulations performed using the Weather Research, and Forecasting (WRF) mesoscale model. To be comparable to ASCAT surface winds, which are referenced at 10 m, the ferry lidar and FINO2 wind profile measurements were extrapolated down to 10 m using atmospheric stability information derived from the bulk Richardson number formulation. ASCAT had the lowest associated error compared with that of the ferry lidar in near-neutral atmospheric stratifications, whereas FINO2, despite a distance range of 30 km and a moving ferry lidar target, had the highest correlation and lowest RMSE in all atmospheric conditions. Due to the high frequency of low-level jets caused by the proximity to land from all directions as well as typically stable atmospheric conditions, the extrapolated ferry lidar measurements underpredicted the ASCAT 10 m wind speeds. WRF consistently underperformed compared to the other measurement methods, even with the ability to directly compare results with all other sources at all heights. Full article

The demand for energy due to the population boom, together with the harmful consequences of fossil fuels, makes it essential to explore renewable thermal energy. Solar Thermal Systems (STS’s) are important alternatives to conventional fossil fuels, owing to their ability to convert solar thermal energy into heat and electricity. However, improving the efficiency of solar thermal systems is the biggest challenge for researchers. Nanomaterial is an effective technique for improving the efficiency of STS’s by using nanomaterials as working fluids. Therefore, the present theoretical study aims to explore the thermal energy characteristics of the flow of nanomaterials generated by the surface gradient (Marangoni convection) on a disk surface subjected to two different thermal energy modulations. Instead of the conventional Fourier heat flux law to examine heat transfer characteristics, the Cattaneo–Christov heat flux (Fourier’s heat flux model) law is accounted for. The inhomogeneous nanomaterial model is used in mathematical modeling. The exponential form of thermal energy modulations is incorporated. The finite-difference technique along with Richardson extrapolation is used to treat the governing problem. The effects of the key parameters on flow distributions were analyzed in detail. Numerical calculations were performed to obtain correlations giving the reduced Nusselt number and the reduced Sherwood number in terms of relevant key parameters. The heat transfer rate of solar collectors increases due to the Marangoni convection. The thermophoresis phenomenon and chaotic movement of nanoparticles in a working fluid of solar collectors enhance the temperature distribution of the system. Furthermore, the thermal field is enhanced due to the thermal energy modulations. The results find applications in solar thermal exchanger manufacturing processes. Full article

This paper proposes an extrapolation method to solve a class of non-linear weakly singular kernel Volterra integral equations with vanishing delay. After the existence and uniqueness of the solution to the original equation are proved, we combine an improved trapezoidal quadrature formula with an interpolation technique to obtain an approximate equation, and then we enhance the error accuracy of the approximate solution using the Richardson extrapolation, on the basis of the asymptotic error expansion. Simultaneously, a posteriori error estimate for the method is derived. Some illustrative examples demonstrating the efficiency of the method are given. Full article

This paper presents a computational study on bottom gas injection in a cylindrical tank. The bubble formation at submerged orifices, bubble rising, and interactions between bubbles and bubbles with the free surface were studied using the conservative level set method (CLSM). Since the gas injection is an important technique in various fields and this process is quite complicated, the scenario was chosen to quantify the efficacy of the CLSM to describe the gas-liquid complex interactions with fast changes in the surface tension force and buoyancy force. The simulation accuracy is verified with the grid convergence index (GCI) approach and Richardson Extrapolation (RE) and is validated by comparing the numerical results with experimental observations, theoretical equations, and published data. The results show that the CLSM accurately reproduces the bubble formation frequency, and that it can handle complicated bubble shapes. Moreover, it captures the challenging phenomena of interaction between bubbles and free surface, the jet of liquid produced when bubbles break through the free surface, and the rupture of the film of liquid. Therefore, the CLSM is a robust numerical technique to describe gas-liquid complex interactions, and it is suited to simulate the gas injection operation. Full article

For reliability and thermal management of power devices, the most frequently used technique is to employ heatsinks. In this work, a new configuration of offset strip fin heatsink based on using the concept of curvy fins and U-turn is proposed with the aim of improving the heat transfer performance. With this aim, a three-dimensional model of heatsink with Silicon Insulated-Gate Bipolar Transistors (IGBTs) and diodes, solder, Direct Bonded Copper (DBC) substrate, baseplate and thermal grease is developed. Richardson’s extrapolation is used for increasing the accuracy of the numerical simulations and to validate the simulations. To study the effectiveness of the new offset design, results are compared with conventional offset strip fin heatsink. Results show that in aspects of design of heatsinks (including heat transfer coefficient, maximum chip temperature and thermal resistance), the new introduced model has advantages compared to the conventional offset strip fin design. These enhancements are caused by the combination of the longer coolant passage in the heatsink associated with generation of disturbance and recirculation areas along the curvy fins, creation of centrifugal forces in the U-turn, and periodic breaking up boundary layers. Also, it is shown that due to narrower passage and back-and-forth route, the new introduced design can handle the hot spots better than conventional design. Full article

During the last decades, the Nonequilibrium Green’s function (NEGF) formalism has been proposed to develop nano-scaled device-simulation tools since it is especially convenient to deal with open device systems on a quantum-mechanical base and allows the treatment of inelastic scattering. In particular, it is able to account for inelastic effects on the electronic and thermal current, originating from the interactions of electron–phonon and phonon–phonon, respectively. However, the treatment of inelastic mechanisms within the NEGF framework usually relies on a numerically expensive scheme, implementing the self-consistent Born approximation (SCBA). In this article, we review an alternative approach, the so-called Lowest Order Approximation (LOA), which is realized by a rescaling technique and coupled with Padé approximants, to efficiently model inelastic scattering in nanostructures. Its main advantage is to provide a numerically efficient and physically meaningful quantum treatment of scattering processes. This approach is successfully applied to the three-dimensional (3D) atomistic quantum transport OMEN code to study the impact of electron–phonon and anharmonic phonon–phonon scattering in nanowire field-effect transistors. A reduction of the computational time by about ×6 for the electronic current and ×2 for the thermal current calculation is obtained. We also review the possibility to apply the first-order Richardson extrapolation to the Padé N/N − 1 sequence in order to accelerate the convergence of divergent LOA series. More in general, the reviewed approach shows the potentiality to significantly and systematically lighten the computational burden associated to the atomistic quantum simulations of dissipative transport in realistic 3D systems. Full article

The study of laminar flow of heat and mass transfer over a moving surface in bionanofluid is of considerable interest because of its importance for industrial and technological processes such as fabrication of bio-nano materials and thermally enhanced media for bio-inspired fuel cells. Hence, the present work deals with the unsteady bionanofluid flow, heat and mass transfer past an impermeable stretching/shrinking sheet. The appropriate similarity solutions transform the boundary layer equations with three independent variables to a system of ordinary differential equations with one independent variable. The finite difference coupled with the Richardson extrapolation technique in the Maple software solves the reduced system, numerically. The rate of heat transfer is found to be higher when the flow is decelerated past a stretching sheet. It is understood that the state of shrinking sheet limits the rate of heat transfer and the density of the motile microorganisms in the stagnation region. Full article

This article introduces a novel approach for generating low-sensitive Pareto fronts of analog circuit performances. The main idea consists of taking advantage from the social interaction between particles within a multi-objective particle swarm optimization algorithm by progressively guiding the global leading process towards low sensitive solutions inside the landscape. We show that the proposed approach significantly outperforms already proposed techniques dealing with the generation of sensitivity-aware Pareto fronts, not only in terms of computing time, but also with regards to the number of solutions forming the tradeoff surface. Performances of our approach are highlighted via the design of two analog circuits. Full article

As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is that it is unconditionally stable and convergent on the order $O({\tau }^2+h^4)$ (where $\tau$ is the time step size and h is the mesh size), under the maximum norm for a class of nonlinear delay partial differential equations with initial and Dirichlet boundary conditions. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. The convergence order of this kind of method is also $O({\tau }^2+h^4)$ under the discrete maximum norm when the spatial step size is twice the one of H-OCD, which accelerates the computational process. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in the time direction. The results of numerical experiments are consistent with the theoretical analysis. Full article