Order Accurate Scheme Research Articles

Analysis of a meshless generalized finite difference method for the time-fractional diffusion-wave equation

In this paper, a generalized finite difference method (GFDM) is proposed and analyzed for meshless numerical solution of the time-fractional diffusion-wave equation. Two (3−α)-order accurate temporal discretization schemes are presented by using the L1 formula and the original H2N2 or fast H2N2 formulas to discretize the time-fractional derivative of order α∈(1,2). The stability of the temporal discretization schemes is analyzed. Then, the time-fractional diffusion-wave initial-boundary value problem is transformed into a series of time-independent integer-order boundary value problems, and discrete linear algebraic systems are built by the application of the GFDM. Accuracy analysis of the GFDM with both original H2N2 and fast H2N2 formulas is presented in theory, and numerical experimental results are provided to verify the theoretical results and the effectiveness of the proposed meshless method.

DNS vs. LES: Turbulent flow in square duct with heated foil boundary

Turbulent heat transfer in a square duct, where a finite section of a single duct boundary is heated with an infinitely thin foil, was analyzed with LES and DNS approaches. Both computations were performed with OpenFOAM code with passive scalar approximation, using second order accurate schemes. Results are obtained for the key velocity and temperature statistics in the duct, with an emphasis on the temperature profiles and temperature fluctuations at the foil boundary, which appear due to the imposed constant heat flux boundary condition at the foil. Comparison of DNS and standard LES-WALE model show 20% higher foil temperatures and temperature fluctuations in LES results. These differences are explained with detailed analysis of the LES-WALE model near-wall behavior. It has been demonstrated that LES-WALE yields highly accurate results only when employed with a finely resolved mesh. However, the computational costs associated with such refinement may not be necessarily justified especially with respect to DNS.

Numerical study of transitions in lid-driven flows in shallow cavities

In this article, three dimensional (3D) lid-driven flow in shallow cavities with a unit square base are studied. The numerical solution of the Navier–Stokes equations modeling incompressible viscous fluid flow in a cavity is obtained via a methodology combining a first order accurate operator-splitting scheme, a L 2 -projection Stokes solver, a wave-like equation treatment of the advection and finite element space approximations. Numerical results of a lid-driven flow in a cubic cavity show a good agreement with those reported in literature. The critical Reynolds numbers (Re cr ) for having flow with increasing of oscillating amplitude (a Hopf bifurcation) in different shallow cavities are obtained and associated oscillating modes are studied.

Fluid Flow Dynamics in Partially Saturated Paper.

This study presents an integrated approach to understanding fluid dynamics in Microfluidic Paper-Based Analytical Devices (µPADs), combining empirical investigations with advanced numerical modeling. Paper-based devices are recognized for their low cost, portability, and simplicity and are increasingly applied in health, environmental monitoring, and food quality analysis. However, challenges such as lack of flow control and the need for advanced detection methods have limited their widespread adoption. To address these challenges, our study introduces a novel numerical model that incorporates factors such as pore size, fiber orientation, and porosity, thus providing a comprehensive understanding of fluid dynamics across various saturation levels of paper. Empirical results focused on observing the wetted length in saturated paper substrates. The numerical model, integrating the Highly Simplified Marker and Cell (HSMAC) method and the High Order accuracy scheme Reducing Numerical Error Terms (HORNET) scheme, successfully predicts fluid flow in scenarios challenging for empirical observation, especially at high saturation levels. The model effectively mimicked the Lucas-Washburn relation for dry paper and demonstrated the increasing time requirement for fluid movement with rising saturation levels. It also accurately predicted faster fluid flow in Whatman Grade 4 filter paper compared with Grade 41 due to its larger pore size and forecasted an increased flow rate in the machine direction fiber orientation of Whatman Grade 4. These findings have significant implications for the design and application of µPADs, emphasizing the need for precise control of fluid flow and the consideration of substrate microstructural properties. The study's combination of empirical data and advanced numerical modeling marks a considerable advancement in paper-based microfluidics, offering robust frameworks for future development and optimization of paper-based assays.

High order accuracy scheme for modeling the dynamics of predator and prey in heterogeneous environment

The aim of this work is to develop a compact finite-difference approach for modeling the dynamics of predator and prey based on reaction-diffusion-advection equations with variable coefficients. Methods. To discretize a spatially inhomogeneous problem with nonlinear terms of taxis and local interaction, the balance method is used. Species densities are determined on the main grid whereas fluxes are computed at the nodes of the staggered grid. Integration over time is carried out using the high-order Runge-Kutta method. Results. For the case of one-dimensional annular interval, the finite-difference scheme on the three-point stencil has been constructed that makes it possible to increase the order of accuracy compared to the standard second-order approximation scheme. The results of computational experiment are presented and comparison of schemes for stationary and non-stationary solutions is carried out. We conduct the calculation of accuracy order basing on the Aitken process for sequences of spatial grids. The calculated values of the effective order accuracy for the proposed scheme were greater than the standard two: for the diffusion problem, values of at least four were obtained. Decrease was obtained when directional migration was taken into account. This conclusion was also confirmed for non-stationary oscillatory regimes. Conclusion. The results demonstrate the effectiveness of the derived scheme for dynamics of predator and prey system in a heterogeneous environment.

Gas composition tracking feasibility using transient finite difference [formula omitted]-scheme model for binary gas mixtures

Implicit finite difference method is popular numerical method for simulation of transient pipeline dynamics, governed by the system of Euler partial differential equations. If advection equation is included in the mathematical formulation, gas composition tracking is also possible. If implicit scheme is used for gas composition tracking, simulated concentration profiles are distorted due to the occurrence of numerical diffusion, a non-physical phenomenon, caused by the 1st order accurate numerical implicit scheme. Intensity of the numerical diffusion can be decreased by use of θ-scheme with arbitrarily chosen implicitness. In order to evaluate accuracy of such numerical scheme and adequacy for usage in actual application, numerical model must be evaluated on actual gas pipeline system topology with actual measured boundary conditions and compared to actual measured results. Aim of this paper is to assess whether hydrogen concentration tracking is feasible using such approach considering complex grid topologies. Paper highlights the peculiarities of the proposed θ-scheme and finite difference method and its impact on the calculated concentration profile for binary gas mixtures. Non-physical flow reversal issue and its impacts on concentration tracking in looped pipeline system is researched and explained. Simulation results shows high accuracy for simulated values of hydraulic variables compared to the measured values, in some cases fit is perfect. Discrepancy between measured and the simulated values of the pressures at nodes on the main transport route is negligible, while the discrepancy increases with the distance from the main transport routes. The largest discrepancy occurred on the most distant node and was between 1.7006 bar (4.85%) and 1.7119 bar (4.88%), depending on the θ value. On the other hand, simulated concentration profile lacks finer details due to the numerical diffusion effects while the simulated concentration profile lags the measured profile, partly by the impacts of the 1st order accurate scheme and partly by effects of non-physical reversed flows. Discrepancy between simulated and measured amplitude of the tracked concentration profile was negligible, while discrepancy due to the time delay of the simulated and measured profile was more expressed and ranged between 2h to 9h. Largest time delay was caused by the non-physical flow reversal.

Fully-discrete spatial eigenanalysis of discontinuous spectral element methods: Insights into well-resolved and under-resolved vortical flows

This study presents a comprehensive spatial eigenanalysis of fully-discrete discontinuous spectral element methods, now generalising previous spatial eigenanalysis that did not include time integration errors. The influence of discrete time integration is discussed in detail for different explicit Runge–Kutta (1st to 4th order accurate) schemes combined with either Discontinuous Galerkin (DG) or Spectral Difference (SD) methods, both here recovered from the Flux Reconstruction (FR) scheme. Selected numerical experiments using the improved SD method by Liang et al. (2009) [53,54] and Jameson (2010) [55] are performed to quantify the influence of time integration errors on actual simulations. These involve test cases of varied complexity, from one-dimensional linear advection equation studies to well-resolved and under-resolved inviscid vortical flows. When simulations are well-resolved, the overall order of accuracy of the (fully-discrete) method of choice is limited to that of the time integration scheme. Moreover, it is shown that, while both well-resolved and under-resolved simulations of linear problems correlate well with the eigenanalysis prediction of time integration errors, the correlation can be much worse for under-resolved nonlinear problems as observed via numerical experiments. In fact, in the numerical simulation of under-resolved vortical flows, the predominance of spatial errors made it practically impossible for time integration errors to be distinctly identified. As a result, the eigenanalysis predictions are expected to hold (even if partially) in direct numerical simulations of turbulence. This highlights that the interaction between space and time discretisation errors is more complex than otherwise anticipated, contributing to the current understanding about when eigenanalysis can effectively predict the behaviour of numerical errors in practical under-resolved nonlinear problems, including under-resolved turbulence computations.

Non-uniform Compact Schemes for Parallel Computing

Here, the performance of a sixth order accurate Non-Uniform Compact Scheme (NUCS6) in a parallel framework using the domain decomposition method is analyzed. Traditional compact schemes are used for uniform grids. However, study of flows involving complicated geometries requires transformation of computational domain onto a uniform grid. Compact schemes developed for physical plane computations eliminate the need for such grid transformations. The first successful implementation of uniform compact schemes on a parallelized framework with Schwartz domain decomposition demonstrated the effect of overlap points in eliminating spurious signals at inter-processor boundaries Sengupta et al. [Jour. Comp. Phy. 220 (2007)]. This paper presents the performance of a high order non-uniform compact scheme (NUCS6) in parallel computations with the domain decomposition method. The results of the benchmark problem of flow in a 2D square lid-driven cavity at post-Hopf bifurcation Reynolds number of 10000 is presented to show how the width of overlap region affects the growth and evolution of gyrating polygonal vortex at the centre of the cavity.

On the Convergence of α Schemes with Source Terms for Scalar Convex Conservation Laws

In this study, we use an extension of Yang’s convergence criterion [N. Jiang, On the wavewise entropy inequality for high-resolution schemes with source terms II: the fully discrete case] to show the entropy convergence of a class of fully discrete α schemes, now with source terms, for non-homogeneous scalar convex conservation laws in the one-dimensional case. The homogeneous counterparts (HCPs) of these schemes were constructed by S. Osher and S. Chakravarthy in the mid-1980s [A New Class of High Accuracy TVD Schemes for Hyperbolic Conservation Laws (1985), Very High Order Accurate TVD Schemes (1986)], and the entropy convergence of these methods, when m=2, was settled by the author [N. Jiang, The Convergence of α Schemes for Conservation Laws II: Fully-Discrete]. For semi-discrete α schemes, with or without source terms, the entropy convergence of these schemes was previously established (for m=2) by the author [N. Jiang, The Convergence of α Schemes for Conservation Laws I: Semi-Discrete Case].

A new family of semi-implicit Finite Volume/Virtual Element methods for incompressible flows on unstructured meshes

We introduce a new family of high order accurate semi-implicit schemes for the solution of nonlinear time-dependent systems of partial differential equations (PDE) on unstructured polygonal meshes. The time discretization is based on a splitting between explicit and implicit terms that may arise either from the multi-scale nature of the governing equations, which involve both slow and fast scales, or in the context of projection methods, where the numerical solution is projected onto the physically meaningful solution manifold. We propose to use a high order finite volume (FV) scheme for the explicit terms, hence ensuring conservation property and robustness across shock waves, while the virtual element method (VEM) is employed to deal with the discretization of the implicit terms, which typically requires an elliptic problem to be solved. The numerical solution is then transferred via suitable L2 projection operators from the FV to the VEM solution space and vice-versa. High order time accuracy is then achieved using the semi-implicit IMEX Runge–Kutta schemes, and the novel schemes are proven to be asymptotic preserving (AP) and well-balanced (WB). As representative models, we choose the shallow water equations (SWE), thus handling multiple time scales characterized by a different Froude number, and the incompressible Navier–Stokes equations (INS), which are solved at the aid of a projection method to satisfy the solenoidal constraint of the velocity field. Furthermore, an implicit discretization for the viscous terms is also devised for the INS model, which is based on the VEM technique. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the fluid velocity and not on the celerity nor on the viscous eigenvalues. A large suite of test cases demonstrates the accuracy and the capabilities of the new family of schemes to solve relevant benchmarks in the field of incompressible fluids.

High order accurate and convergent numerical scheme for the strongly anisotropic Cahn–Hilliard model

AbstractWe propose and analyze a second order accurate in time, energy stable numerical scheme for the strongly anisotropic Cahn–Hilliard system, in which a biharmonic regularization has to be introduced to make the equation well‐posed. A convexity analysis on the anisotropic interfacial energy is necessary to overcome an essential difficulty associated with its highly nonlinear and singular nature. The second order backward differentiation formula temporal approximation is applied, combined with Fourier pseudo‐spectral spatial discretization. The nonlinear surface energy part is updated by an explicit extrapolation formula. Meanwhile, the energy stability analysis is enforced by the fact that all the second order functional derivatives of the energy stay uniformly bounded by a global constant. A Douglas‐Dupont type regularization is added to stabilize the numerical scheme, and a careful estimate ensures a modified energy stability with a uniform constraint for the regularization parameter . In turn, the combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear scheme. More importantly, such an energy stability is in terms of the interfacial energy with respect to the original phase variable, which enables us to derive an optimal rate convergence analysis.

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

Analysis of Heat Transport in a Powell-Eyring Fluid with Radiation and Joule Heating Effects via a Similarity Transformation

Heat transfer in an Eyring-Powell fluid that conducts electricity and flows past an exponentially growing sheet is considered. As the sheet is stretched in the x direction, the flow develops in the region with y > 0. The problem is tackled through a set of partial differential equations accounting for Magnetohydrodynamics (MHD), radiation and Joule heating effects, which are converted into a set of equivalent ordinary differential equations through a similarity transformation. The converted problem is solved in MATLAB in the framework a fourth order accurate integration scheme. It is found that the thermal relaxation period is inversely proportional to the thickness of the thermal boundary layer, whereas the Eckert-number displays the opposite trend. As this characteristic number grows, the temperature within the channel increases.

High resolution compact implicit numerical scheme for conservation laws

We present a novel implicit scheme for the numerical solution of time-dependent conservation laws. The core idea of the presented method is to exploit and approximate the mixed spatial-temporal derivative of the solution that occurs naturally when deriving some second order accurate schemes in time. Such an approach is introduced in the context of the Lax-Wendroff (or Cauchy-Kowalevski) procedure when the second time derivative is not completely replaced by space derivatives using the PDE, but the mixed derivative is kept. If approximated in a suitable way, the resulting compact implicit scheme produces algebraic systems that have a more convenient structure than the systems derived by fully implicit schemes. We derive a high resolution TVD form of the implicit scheme for some representative hyperbolic equations in the one-dimensional case, including illustrative numerical experiments.

Numerical investigation of low-frequency instability and frequency shifting in a scramjet combustor

The combustion instability in a laboratory-scale direct-connect hydrogen-fueled scramjet combustor is investigated numerically. The numerical simulation has been carried out using a delayed detached eddy simulation (DDES) with a detailed reaction mechanism. The computational framework has high fidelity by applying multi-dimensional high order accurate schemes for handling convective and viscous fluxes. The field data were accumulated up to 100 milliseconds on each case to capture sufficiently the repetitive behavior of low-frequency instability of order of 100 Hz. The numerical results exhibit the formation/dissipation of pressure and shock wave induced by continuous heat release in the combustor. This motion of pressure/shock wave, so-called upstream-traveling shock wave, presents repeated dynamics between isolator and combustor with a period of several milliseconds. With this periodic hydrodynamic characteristic, the upstream-traveling shock wave interacts with the boundary layer and injected fuel stream affecting fuel/air mixing and burning, and finally inducing the combustion instability in a scramjet combustor. Frequency analysis derived major instability frequencies of 190 Hz and 450 Hz in the isolator and combustor for low and high equivalence ratios, respectively. Current numerical results present the underlying flow physics on the shifting of the instability frequency by changing the equivalence ratio observed by the previous experimental studies. The fact that an instability frequency exists homogeneously from isolator to combustor informs that the combustion instability of scramjet engine is the fully coupled flow/combustion dynamics throughout the engine on a macroscopic scale.

3D time-dependent scattering about complex shapes using high order difference potentials

We compute the scattering of unsteady acoustic waves about complex three-dimensional bodies with high order accuracy. The geometry of a scattering body is defined with the help of CAD. Its surface is represented as a collection of non-overlapping patches, each parameterized independently by means of high order splines (NURBS). As a specific example, we consider a submarine-like scatterer constructed using five different patches.The acoustic wave equation on the region exterior to the scatterer is solved by first reducing it to a system of Calderon's boundary operator equations. The latter are obtained using the method of difference potentials coupled with a compact fourth order accurate finite difference scheme. When solving the boundary operator equations, we employ Huygens' principle. It allows us to work on a sliding time window of non-increasing duration rather than keep the entire temporal history of the solution at the boundary.The proposed methodology demonstrates grid-independent computational complexity at the boundary and sub-linear complexity with respect to the grid dimension. It efficiently handles complex non-conforming geometries on Cartesian grids with no penalty for either accuracy or stability due to the cut cells. Its performance does not deteriorate over arbitrarily long simulation times. The exact treatment of artificial outer boundaries is inherently built in. Finally, multiple similar problems can be solved efficiently at a low individual cost per problem. This is important when, for example, the boundary condition on the surface changes but the scattering body stays the same.

A Monotone, Second Order Accurate Scheme for Curvature Motion

We present a second order accurate in time numerical scheme for curve shortening flow in the plane that is unconditionally monotone. It is a variant of threshold dynamics, a class of algorithms in the spirit of the level set method that represent interfaces implicitly. The novelty is monotonicity: it is possible to preserve the comparison principle of the exact evolution while achieving second order in time consistency. As a consequence of monotonicity, convergence to the viscosity solution of curve shortening is ensured by existing theory.

Implicit-explicit relaxation Runge-Kutta methods: construction, analysis and applications to PDEs

Spatial discretizations of time-dependent partial differential equations usually result in a large system of semi-linear and stiff ordinary differential equations. Taking the structures into account, we develop a family of linearly implicit and high order accurate schemes for the time discretization, using the idea of implicit-explicit Runge-Kutta methods and the relaxation techniques. The proposed schemes are monotonicity-preserving/conservative for the original problems, while the previous linearized methods are usually not. We also discuss the linear stability and strong stability preserving (SSP) property of the new relaxation methods. Numerical experiments on several typical models are presented to confirm the effectiveness of the proposed methods.

A high order compact time/space finite difference scheme for the 2D and 3D wave equation with a damping layer

We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. For unbounded domains we add a fourth order accurate sponge layer to damp the outgoing waves. We demonstrate that usually this is more efficient than lower order schemes despite being implicit and conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as multi-grid. Computations confirm the design convergence rate for the in-homogeneous, variable wave speed equation.

Discrete maximum principle of a high order finite difference scheme for a generalized Allen–Cahn equation

We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation of the $Q^2$ spectral element method. We prove that the discrete maximum principle holds under suitable mesh size and time step constraints. The same result also applies to construct a bound-preserving scheme for any passive convection with an incompressible velocity field.