Search

Any robust computational scheme for compressible flows must retain the hyperbolicity property or the real‐valued sound speed. Failure to maintain hyperbolicity, or the positivity of the square of the speed of sound, causes nonphysical distortions and the blow‐up of numerical simulations. Strong shock waves and interfacial discontinuities are ubiquitous features of the two‐fluid compressible dynamics that can potentially induce positivity‐related failure in a simulation. This article presents a positivity‐preserving algorithm for a high‐order, primitive variable‐based, weighted essentially non‐oscillatory finite difference scheme. The positivity preservation relies on a flux limiting technique that locally adapts high‐order fluxes towards the first order to retain the physical bounds of the solution without loss of high‐order convergence. This positivity preserving scheme has been devised and implemented up to eleventh order in one and two dimensions for a two‐fluid compressible model that consists of a single mass, momentum, and energy equations, as well as an advection of material parameters for capturing the interfaces. Several one‐ and two‐dimensional challenging test problems verify the performance. The scheme effectively retains high order accuracy while allowing for the simulation of several challenging problems that otherwise could not be successfully solved using the base scheme, without any penalty on the CFL condition requirement, and without any significant impact on the CPU times. The scheme represents the first high‐order (up to 11‐order) hyperbolicity‐preserving scheme for the considered two‐fluid compressible flows in the fully Eulerian formulation. The inherent efficiency of finite differences and the new robust positivity preserving quality enable modeling other challenging problems of two‐fluid and two‐phase problems. [ABSTRACT FROM AUTHOR]

This paper focuses on the development of an explicit finite difference numerical method for approximating the solution of the inhomogeneous fourth-order Euler–Bernoulli beam bending equation with velocity-dependent damping and second moment of area, mass and elastic modulus distribution varying with distance along the beam. We verify the method by comparing its predictions with an exact analytical solution of the homogeneous equation, we use the generalised Richardson extrapolation to show that the method is grid convergent and we extend the application of the Lax–Richtmyer stability criteria to higher-order schemes to ensure that it is numerically stable. Finally, we present three sets of computational experiments. The first set simulates the behaviour of the un-loaded beam and is validated against the analytic solution. The second set simulates the time-dependent dynamic behaviour of a damped beam of varying stiffness and mass distributions under arbitrary externally applied loading in an aeroelastic analysis setting by approximating the inhomogeneous equation using the finite difference method derived here. We compare the third set of simulations of the steady-state deflection with the results of static beam bending experiments conducted at Cranfield University. Overall, we developed an accurate, stable and convergent numerical framework for solving the inhomogeneous Euler–Bernoulli equation over a wide range of boundary conditions. Aircraft manufacturers are starting to consider configurations with increased wing aspect ratios and reduced structural weight which lead to more slender and flexible designs. Aeroelastic analysis now plays a central role in the design process. Efficient computational tools for the prediction of the deformation of wings under external loads are in demand and this has motivated the work carried out in this paper.

Two- and three-dimensional direct numerical simulation (DNS) of turbine cascade flow at low Reynolds number with heat transfer are performed using high-order finite difference method. Two-dimensional laminar computation which is used to construct the initial flow of three-dimensional DNS fails to predict Stanton number on the second half of suction side where the flow is turbulent in experiment. In three-dimensional DNS, transition is triggered by periodic blow-and-suction disturbances. Numerical experiments show that phase randomness of the disturbance is not necessary to trigger the transition which can be induced by disturbances with fixed phases. In a range of time fundamental frequency of disturbance, when increasing the frequency, transition moves downstream. When time fundamental frequency is big enough, transition disappears. With fixed space phases, time phases and selected time fundamental frequency, time averaged pressure and Stanton number distributions of three-dimensional DNS coincide with the experimental datum. Averaged velocity and temperature, Root-Mean-squares (RMS) of velocity pulse，temperature pulse, Reynolds shear stress and heat flux are extracted from the DNS database. All statistics agree well with experimental and theoretical results which verify the accuracy of present database. [ABSTRACT FROM AUTHOR]

In this paper, we study and analyze Crank-Nicolson temporal discretization with high-order spatial difference schemes for time-dependent Riesz space-fractional diffusion equations with variable diffusion coefficients. To the best of our knowledge, there is no stability and convergence analysis for temporally 2nd-order or spatially jth-order (j≥3) difference schemes for such equations with variable coefficients. We prove under mild assumptions on diffusion coefficients and spatial discretization schemes that the resulting discretized systems are unconditionally stable and convergent with respect to discrete ℓ2-norm. We further show that several spatial difference schemes with jth-order (j=1,2,3,4) truncation error satisfy the assumptions required in our analysis. As a result, we obtain a series of temporally 2nd-order and spatially jth-order (j=1,2,3,4) unconditionally stable difference schemes for solving time-dependent Riesz space-fractional diffusion equations with variable coefficients. Numerical results are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Many phenomena in Physics and Engineering are modeled by Partial Differential Equations. Typically, analytical solutions of such problems are unknown. Hence, numerical schemes are applied for approximating the exact solutions. In order to accurately approximate the exact solutions, very fine grids are required. However, the computations become too costly. In this work, a multiscale iterative approach is utilized for enhancing the accuracy of coarse grid computations which results from a high-order finite difference scheme. The main idea is to run the finite difference computations on a coarse grid until some intermediate time level. Then, the coarse grid results are interpolated and extended to a fine grid using Laplacian Pyramids, a multiscale iterative approach. Finally, the finite difference scheme is employed on the finer grid until a prescribed final time. Comparing the results to those obtained on a fine grid, we see that the convergence rates obtained from the two methods are comparable, while the computational time is significantly reduced. A modified multimodal Laplacian Pyramids algorithm for predicting future values of the solution is also suggested. The method approximates and extends a function based on two or more input modalities coded by a series of multiscale kernels, which are averaged as a convex combination. In this work, the modalities are the numerical model's approximations of the solution of the differential equation and its derivative at previous time steps, and the goal is to predict the solution at a proceeding time step. It can be seen that, by adapting the convex combination of the kernels to local regions of the solution with stronger or weaker gradients, the predicted results are improved. [ABSTRACT FROM AUTHOR]

Highly-accurate numerical methods that can efficiently handle problems with interfaces and/or problems in domains with complex geometry are crucial for the resolution of different temporal and spatial scales in many problems from physics and biology. In this paper we continue the work started in [8] , and we use modest one-dimensional parabolic problems as the initial step towards the development of high-order accurate methods based on the Difference Potentials approach. The designed methods are well-suited for variable coefficient parabolic models in heterogeneous media and/or models with non-matching interfaces and with non-matching grids. Numerical experiments are provided to illustrate high-order accuracy and efficiency of the developed schemes. While the method and analysis are simpler in the one-dimensional settings, they illustrate and test several important ideas and capabilities of the developed approach. [ABSTRACT FROM AUTHOR]

Numerical approximations and modeling of many physical, biological, and biomedical problems often deal with equations with highly varying coefficients, heterogeneous models (described by different types of partial differential equations (PDEs) in different domains), and/or have to take into consideration the complex structure of the computational subdomains. The major challenge here is to design an efficient numerical method that can capture certain properties of analytical solutions in different domains/subdomains (such as positivity, different regularity/smoothness of the solutions, etc.), while handling the arbitrary geometries and complex structures of the domains. In this work, we employ one-dimensional elliptic type models as the starting point to develop and numerically test high-order accurate Difference Potentials Method (DPM) for variable coefficient elliptic problems in heterogeneous media. While the method and analysis are simple in the one-dimensional settings, they illustrate and test several important ideas and capabilities of the developed approach. [ABSTRACT FROM AUTHOR]

Abstract: The geometric conservation law (GCL) includes the volume conservation law (VCL) and the surface conservation law (SCL). Though the VCL is widely discussed for time-depending grids, in the cases of stationary grids the SCL also works as a very important role for high-order accurate numerical simulations. The SCL is usually not satisfied on discretized grid meshes because of discretization errors, and the violation of the SCL can lead to numerical instabilities especially when high-order schemes are applied. In order to fulfill the SCL in high-order finite difference schemes, a conservative metric method (CMM) is presented. This method is achieved by computing grid metric derivatives through a conservative form with the same scheme applied for fluxes. The CMM is proven to be a sufficient condition for the SCL, and can ensure the SCL for interior schemes as well as boundary and near boundary schemes. Though the first-level difference operators δ 3 have no effects on the SCL, no extra errors can be introduced as δ 3 = δ 2. The generally used high-order finite difference schemes are categorized as central schemes (CS) and upwind schemes (UPW) based on the difference operator δ 1 which are used to solve the governing equations. The CMM can be applied to CS and is difficult to be satisfied by UPW. Thus, it is critical to select the difference operator δ 1 to reduce the SCL-related errors. Numerical tests based on WCNS-E-5 show that the SCL plays a very important role in ensuring free-stream conservation, suppressing numerical oscillations, and enhancing the robustness of the high-order scheme in complex grids. [ABSTRACT FROM AUTHOR]

Abstract: In this work we derive high-order (second and fourth of convergence, but with first and third-order local truncation error, respectively) compact finite difference schemes for elliptic equations with intersecting interfaces. The approach uses the differential equation and the jump (interface) relations as additional identities which can be differentiated to eliminate higher order local truncation errors. Numerical experiments are carried out to demonstrate the high-order accuracy and to show that our method is effective to sharp contrast in the diffusion coefficients of the problems. [Copyright &y& Elsevier]

The direct numerical simulation of receptivity, instability and transition of hypersonic boundary layers requires high-order accurate schemes because lower-order schemes do not have an adequate accuracy level to compute the large range of time and length scales in such flow fields. The main limiting factor in the application of high-order schemes to practical boundary-layer flow problems is the numerical instability of high-order boundary closure schemes on the wall. This paper presents a family of high-order non-uniform grid finite difference schemes with stable boundary closures for the direct numerical simulation of hypersonic boundary-layer transition. By using an appropriate grid stretching, and clustering grid points near the boundary, high-order schemes with stable boundary closures can be obtained. The order of the schemes ranges from first-order at the lowest, to the global spectral collocation method at the highest. The accuracy and stability of the new high-order numerical schemes is tested by numerical simulations of the linear wave equation and two-dimensional incompressible flat plate boundary layer flows. The high-order non-uniform-grid schemes (up to the 11th-order) are subsequently applied for the simulation of the receptivity of a hypersonic boundary layer to free stream disturbances over a blunt leading edge. The steady and unsteady results show that the new high-order schemes are stable and are able to produce high accuracy for computations of the nonlinear two-dimensional Navier–Stokes equations for the wall bounded supersonic flow. [Copyright &y& Elsevier]

This paper focuses on the development of an explicit finite difference numerical method for approximating the solution of the inhomogeneous fourth-order Euler–Bernoulli beam bending equation with velocity-dependent damping and second moment of area, mass and elastic modulus distribution varying with distance along the beam. We verify the method by comparing its predictions with an exact analytical solution of the homogeneous equation, we use the generalised Richardson extrapolation to show that the method is grid convergent and we extend the application of the Lax–Richtmyer stability criteria to higher-order schemes to ensure that it is numerically stable. Finally, we present three sets of computational experiments. The first set simulates the behaviour of the un-loaded beam and is validated against the analytic solution. The second set simulates the time-dependent dynamic behaviour of a damped beam of varying stiffness and mass distributions under arbitrary externally applied loading in an aeroelastic analysis setting by approximating the inhomogeneous equation using the finite difference method derived here. We compare the third set of simulations of the steady-state deflection with the results of static beam bending experiments conducted at Cranfield University. Overall, we developed an accurate, stable and convergent numerical framework for solving the inhomogeneous Euler–Bernoulli equation over a wide range of boundary conditions. Aircraft manufacturers are starting to consider configurations with increased wing aspect ratios and reduced structural weight which lead to more slender and flexible designs. Aeroelastic analysis now plays a central role in the design process. Efficient computational tools for the prediction of the deformation of wings under external loads are in demand and this has motivated the work carried out in this paper. [ABSTRACT FROM AUTHOR]