High-order Numerical Schemes Research Articles

Fourth-order numerical method for the space–time tempered fractional diffusion-wave equation

The main aim of the current paper is to develop a high-order numerical scheme to solve the space–time tempered fractional diffusion-wave equation. The convergence order of the proposed method is O ( τ 2 + h 4 ) . Also, we prove the unconditional stability and convergence of the developed method. The numerical results show the efficiency of the provided numerical scheme.

Deferred Correction Methods for Forward Backward Stochastic Differential Equations

AbstractThe deferred correction (DC) method is a classical method for solving ordinary differential equations; one of its key features is to iteratively use lower order numerical methods so that high-order numerical scheme can be obtained. The main advantage of the DC approach is its simplicity and robustness. In this paper, the DC idea will be adopted to solve forward backward stochastic differential equations (FBSDEs) which have practical importance in many applications. Noted that it is difficult to design high-order and relatively “clean” numerical schemes for FBSDEs due to the involvement of randomness and the coupling of the FSDEs and BSDEs. This paper will describe how to use the simplest Euler method in each DC step–leading to simple computational complexity–to achieve high order rate of convergence.

High Order Numerical Schemes for Second-Order FBSDEs with Applications to Stochastic Optimal Control

AbstractThis is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve the second-order FBSDEs (2FBSDEs). The key feature of the multistep schemes is that the Euler method is used to discretize the forward SDE, which dramatically reduces the entire computational complexity. Moreover, it is shown that the usual quantities of interest (e.g., the solution tuple (Yt,Zt,At,Γt) of the 2FBSDEs) are still of high order accuracy. Several numerical examples are given to show the effectiveness of the proposed numerical schemes. Applications of our numerical schemes to stochastic optimal control problems are also presented.

A new numerical method for variable order fractional functional differential equations

In this letter, a high order numerical scheme is proposed for solving variable order fractional functional differential equations. Firstly, the problem is approximated by an integer order functional differential equation. The integer order differential equation is then solved by the reproducing kernel method. Numerical examples are given to demonstrate the theoretical analysis and verify the efficiency of the proposed method.

Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models

Excitatory and inhibitory nonlinear noisy leaky integrate and fire models are often used to describe neural networks. Recently, new mathematical results have provided a better understanding of them. It has been proved that a fully excitatory network can blow-up in finite time, while a fully inhibitory network has a global in time solution for any initial data. A general description of the steady states of a purely excitatory or inhibitory network has been also given. We extend this study to the system composed of an excitatory population and an inhibitory one. We prove that this system can also blow-up in finite time and analyse its steady states and long time behaviour. Besides, we illustrate our analytical description with some numerical results. The main tools used to reach our aims are: the control of an exponential moment for the blow-up results, a more complicate strategy than that considered in [ 5 ] for studying the number of steady states, entropy methods combined with Poincare inequalities for the long time behaviour and, finally, high order numerical schemes together with parallel computation techniques in order to obtain our numerical results.

Численная схема высокого порядка точности для определения интенсивности вихревого слоя при решении двумерных задач аэрогидродинамики вихревыми методами

Численная схема высокого порядка точности для определения интенсивности вихревого слоя при решении двумерных задач аэрогидродинамики вихревыми методами

Bubble collapse near a fluid-fluid interface using the spectral element marker particle method with applications in bioengineering

The spectral element marker particle (SEMP) method is a high-order numerical scheme for modelling multiphase flow where the governing equations are discretised using the spectral element method and the (compressible) fluid phases are tracked using marker particles. Thus far, the method has been successfully applied to two-phase problems involving the collapse of a two-dimensional bubble in the vicinity of a rigid wall. In this article, the SEMP method is extended to include a third fluid phase before being applied to bubble collapse problems near a fluid-fluid interface. Two-phase bubble collapse near a rigid boundary (where a highly viscous third phase approximates the rigid boundary) is considered as validation of the method. A range of fluid parameter values and geometric configurations are studied before a bioengineering application is considered. A simplified model of (micro)bubble-cell interaction is presented, with the aim of gaining initial insights into the flow mechanisms behind sonoporation and microbubble-enhanced targeted drug delivery. Results from this model indicate that the non-local cell membrane distortion (blebbing) phenomenon often observed experimentally may result from stress propagation along the cell surface and so be hydrodynamical in origin.

Numerical Analysis Using a Fixed Grid Method for Cardiovascular Flow Application

Motivated by the current interest in the numerical simulation of biological flows in the human body, we develop a new method to simulate fluid flow embedded in a solid region. The novelty of this method lies on the use of a fixed grid in the entire computational domain. The formulation is an extension of the multiphase fluid flow that belongs to the category of the penalty method, where high viscosity is imposed on a solid region. A free open source library, namely, OpenFOAM, is used to integrate high order and advanced numerical schemes into these computational formulations. The Monotone Upstream System for Conservation Laws (MUSCL) scheme by van Leer, with a harmonic limiter from the category of the total variation bounded (TVB) scheme, is used for cell face interpolation. The robustness and accuracy of the solver are compared with the benchmark test case, namely, the free fall of a solid sphere. The test case validates that the rigidity of the solid sphere is ensured with the selected high viscosity ratio. The accurate terminal velocity of the falling solid sphere proves the no-slip condition at the solid-liquid interface. As a real application implementation, the flow on a simplified idealized model of heart valve stenosis is presented.

An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations

Current discretizations of variable-order fractional (V-OF) differential equations lead to numerical solutions of low order of accuracy. This paper explores a high order numerical scheme for multi-dimensional V-OF Schrödinger equations. We derive new operational matrices for the V-OF derivatives of Caputo and Riemann–Liouville type of the shifted Jacobi polynomials (SJPs). These allow us to establish an efficient approximate formula for the Riesz fractional derivative. An operational approach of the Jacobi collocation approach for the approximate solution of the V-OF nonlinear Schrödinger equations. The main characteristic behind this approach is to investigate a space–time spectral approximation for spatial and temporal discretizations. The proposed spectral scheme, both in temporal and spatial discretizations, is successfully developed to handle the two-dimensional V-OF Schrödinger equation. Numerical results indicating the spectral accuracy and effectiveness of this algorithm are presented.

A Collocation Boundary Value Method for Linear Volterra Integral Equations

This paper is devoted to studying the boundary value method for Volterra integral equations. High order numerical schemes are devised by using special multistep collocation methods, which depend on numerical approximations of the solution in the next several steps. Stability analysis illustrates these methods enjoy wide absolutely stable regions. With the help of efficient evaluation for highly oscillatory integrals, these methods are applied to solving Volterra integral equations with highly oscillatory kernels. Both theoretical and numerical results show they share the property that the higher the oscillation, the better the accuracy of the approximations.

High-order numerical schemes based on difference potentials for 2D elliptic problems with material interfaces

Numerical approximations and computational modeling of problems from Biology and Materials Science often deal with partial differential equations with varying coefficients and domains with irregular geometry. The challenge here is to design an efficient and accurate numerical method that can resolve properties of solutions in different domains/subdomains, while handling the arbitrary geometries of the domains. In this work, we consider 2D elliptic models with material interfaces and develop efficient high-order accurate methods based on Difference Potentials for such problems.

The Influence of Incoming Boundary Layer on the Flow around the Surface Excrescence

This paper is concerned with the accurate prediction of parasitic drag due to excrescences introduced by structural repairs to aircraft wings and fuselage. Numerical computations are carried out to quantify the effect of isolated surface excrescences in a turbulent boundary layer on a flat plate. Both step excrescence in 2-D domain and wall-mounted hexahedron in 3-D domain are considered in the investigation. Various approaches for calculating drag coefficient increments for wing repair plates are presented. Menter's Shear Stress Transport (SST k-ω) turbulence model is employed here for its accurate prediction of aeronautics flows at typical repair locations with strong adverse pressure gradients and separation. Solutions are obtained via a high-order numerical scheme and an implicit time-marching approach on a multi-block structured mesh. A grid resolution study was carried out to confirm the accuracy of the computations. From the results curves are drawn showing the variation of parasitic drag components for a range of controlled speed (Mach 0.1 to 0.8), local Reynolds number (104 to 107) and the height of excrescence (y+=102 to 104).

1-D 오일러 방정식에 관한 Modal 불연속 갤러킨 기법에서의 Limiter 성능 비교

Considerable efforts are required to develop a monotone, robust and stable high-order numerical scheme for solving the hyperbolic system. The discontinuous Galerkin(DG) method is a natural choice, but elimination of the spurious oscillations from the high-order solutions demands a new development of proper limiters for the DG method. There are several available limiters for controlling or removing unphysical oscillations from the high-order approximate solution; however, very few studies were directed to analyze the exact role of the limiters in the hyperbolic systems. In this study, the performance of the several well-known limiters is examined by comparing the high-order(p 1 , p 2 , and p 3 ) approximate solutions with the exact solutions. It is shown that the accuracy of the limiter is in general problem-dependent, although the Hermite WENO limiter and maximum principle limiter perform better than the TVD and generalized moment limiters for most of the test cases. It is also shown that application of the troubled cell indicators may improve the accuracy of the limiters under some specific conditions.

A high order numerical scheme for variable order fractional ordinary differential equation

In this paper, we derive a high order numerical scheme for variable order fractional ordinary differential equation by establishing a second order numerical approximation to variable order Riemann–Liouville fractional derivative. The scheme is strictly proved to be stable and convergent with second order accuracy, which is higher than some recently derived schemes. Finally, some numerical examples are presented to demonstrate the theoretical analysis and verify the efficiency of the proposed method.

On the convergence of a high-accuracy compact conservative scheme for the modified regularized long-wave equation.

In this article, we develop a high-order efficient numerical scheme to solve the initial-boundary problem of the MRLW equation. The method is based on a combination between the requirement to have a discrete counterpart of the conservation of the physical “energy” of the system and finite difference method. The scheme consists of a fourth-order compact finite difference approximation in space and a version of the leap-frog scheme in time. The unique solvability of numerical solutions is shown. A priori estimate and fourth-order convergence of the finite difference approximate solution are discussed by using discrete energy method and some techniques of matrix theory. Numerical results are given to show the validity and the accuracy of the proposed method.

A high-order local time stepping finite volume solver for one-dimensional blood flow simulations: application to the ADAN model.

In recent years, the complexity of vessel networks for one-dimensional blood flow models has significantly increased, because of enhanced anatomical detail or automatic peripheral vasculature generation, for example. This fact, along with the application of these models in uncertainty quantification and parameter estimation poses the need for extremely efficient numerical solvers. The aim of this work is to present a finite volume solver for one-dimensional blood flow simulations in networks of elastic and viscoelastic vessels, featuring high-order space-time accuracy and local time stepping (LTS). The solver is built on (i) a high-order finite volume type numerical scheme, (ii) a high-order treatment of the numerical solution at internal vertexes of the network, often called junctions, and (iii) an accurate LTS strategy. The accuracy of the proposed methodology is verified by empirical convergence tests. Then, the resulting LTS scheme is applied to arterial networks of increasing complexity and spatial scale heterogeneity, with a number of one-dimensional segments ranging from a few tens up to several thousands and vessel lengths ranging from less than a millimeter up to tens of centimeters, in order to evaluate its computational cost efficiency. The proposed methodology can be extended to any other hyperbolic system for which network applications are relevant. Copyright © 2016 John Wiley & Sons, Ltd.

Performance assessment of low-order versus high-order numerical schemes in the numerical simulation of aquifer flow

Numerical methods have been widely used to simulate transient groundwater flow induced by pumping wells in geometrically and mathematically complex systems. However, flow and transport simulation using low-order numerical methods can be computationally expensive with a low rate of convergence in multi-scale problems where fine spatial discretization is required to ensure stability and desirable accuracy (for instance, close to a pumping well). Numerical approaches based on high-order test functions may better emulate the global behavior of parabolic and/or elliptic groundwater governing equations with and without the presence of pumping well(s). Here, we assess the appropriateness of high-order differential quadrature method (DQM) and radial basis function (RBF)-DQM approaches compared to low-order finite difference and finite element methods. This assessment is carried out using the exact analytical solution by Theis and observed head data as benchmarks. Numerical results show that high-order DQM and RBF-DQM are more efficient schemes compared to low-order numerical methods in the simulation of 1-D axisymmetric transient flow induced by a pumping well. Mesh-less RBF-DQM, with the ability to implement arbitrary (e.g., adaptive) node distribution, properly simulates 2-D transient flow induced by pumping wells in confined/unconfined aquifers with regular and irregular geometries, compared to the other high-order and low-order approaches presented in this paper.

Large Eddy Simulations (LES) including Conditional Source-term Estimation (CSE) applied to two Delft-Jet-in-Hot-Coflow (DJHC) flames

For the first time, a double conditioned CSE formulation including two mixture fractions is applied to model two Delft-Jet-in-Hot-Coflow (DJHC) flames in Large Eddy Simulation (LES). The objective is to further determine the ability of CSE to accurately predict the flame characteristics seen in the Dutch natural gas DJHC flames in the context of LES. Tabulated detailed chemistry and radiation are included. Good agreement with experimental data is found for the turbulent flow field and temperature statistics. Some discrepancies are noted near the centreline due to inaccuracies in the coflow entrainment. For both flames, the radial velocity component is shown to be more difficult to predict accurately. Possible sources of discrepancy are the sub-grid scale model and the inlet turbulence conditions. The predicted lift-off heights are lower than the experimental values. However, the trend of increasing lift-off height with increasing Reynolds number is properly captured when the same fuel composition is considered in the experiments and LES-CSE. The lift-off height predictions may be improved by adding a third conditioning variable that would reflect the effect of straining or changing the chemistry tabulation using different chemical progress variables. Qualitatively, the present simulations correctly predict flame stabilization due to the formation of ignition kernels. Further improvements may be obtained by including temperature fluctuations at the coflow inlet, a dynamic Smagorinsky model and higher order numerical schemes.

Steady-state laminar flow solutions for NACA 0012 airfoil

In this paper we consider the solution of the compressible Navier–Stokes equations for a class of laminar airfoil flows. The principal objective of this paper is to demonstrate that members of this class of laminar flows have steady-state solutions. These laminar airfoil flow cases are often used to evaluate accuracy, stability and convergence of numerical solution algorithms for the Navier–Stokes equations. In recent years such flows have also been used as test cases for high-order numerical schemes. While generally consistent steady-state solutions have been obtained for these flows using higher order schemes, a number of results have been published with various solutions, including unsteady ones. We demonstrate with two different numerical methods and a range of meshes with a maximum density that exceeds 8 × 106 grid points that steady-state solutions are obtained. Furthermore, numerical evidence is presented that even when solving the equations with an unsteady algorithm, one obtains steady-state solutions.

Adaptive time stepping for vesicle suspensions

We present an adaptive high-order accurate time-stepping numerical scheme for the flow of vesicles suspended in Stokesian fluids. Our scheme can be summarized as an approximate implicit spectral deferred correction (SDC) method. Applying a fully-implicit SDC scheme to vesicle flows is prohibitively expensive. For this reason we introduce several approximations. Our scheme is based on a semi-implicit linearized low-order time stepping method. (Our discretization is spectrally accurate in space.) We expect that the accuracy can be arbitrary-order, but our examples suffer from order reduction which limits the observed accuracy to third-order. We also use invariant properties of vesicle flows, constant area and boundary length in two dimensions, to reduce the computational cost of error estimation for adaptive time stepping. We present results in two dimensions for single-vesicle flows, constricted geometry flows, converging flows, and flows in a Couette apparatus. We experimentally demonstrate that the proposed scheme enables automatic selection of the time step size and high-order accuracy.