High-order Numerical Schemes Research Articles

Subgrid-scale fire front reconstruction for ensemble coupled atmosphere-fire simulations of the FireFlux I experiment

This paper introduces the Blaze fire model based on a Eulerian level-set front-tracking method and solved using high-order numerical schemes. Blaze includes an original and efficient subgrid-scale fire front reconstruction to substantially reduce computational cost and better localize surface heat fluxes compared to a weighted-averaged method. In this study, Blaze is coupled to the MesoNH atmospheric model to evaluate its performance against the FireFlux I experimental data set. Results show good agreement between simulations and measurements for both 25-m and 10-m atmospheric resolutions combined with a 5-m fire resolution. The fire-induced atmospheric flow below 10 m is correctly captured in the two-way coupled mode and leads to a realistic spread rate trend between the two instrumented towers compared to one-way forcing modes (forced and fire replay modes). A more realistic air temperature near the ground is obtained by considering heat fluxes in the already burnt area and not only at the flaming front. Also, the significant impact of inflow turbulence on both fire spread and fire-induced flow is highlighted. This study motivates the use of a statistical ensemble technique to account for near-surface turbulence and more generally, environmental variability at the scale of an experimental fire such as FireFlux I.

A two-dimensional high-order well-balanced scheme for the shallow water equations with topography and Manning friction

We develop a two-dimensional high-order numerical scheme that exactly preserves and captures the moving steady states of the shallow water equations with topography or Manning friction. The high-order accuracy relies on a suitable polynomial reconstruction, while the well-balancedness property is based on the first-order scheme from Michel-Dansac et al. (2016, 2017), extended to two space dimensions. To get both properties, we use a convex combination between the high-order scheme and the first-order well-balanced scheme. By adequately choosing the convex combination parameter following a very simple steady state detector, we ensure that the resulting scheme is both high-order accurate and well-balanced. The method is then supplemented with a MOOD procedure to eliminate the spurious oscillations coming from the high-order polynomial reconstruction and to guarantee the physical admissibility of the solution. Numerical experiments show that the scheme indeed possesses the claimed properties. The simulation of the 2011 Tōhoku tsunami, on real data, further confirms the relevance of this technique.

A superconvergent B-spline technique for second order nonlinear boundary value problems

In the present work, a high-order numerical scheme based on B-spline functions is developed for solving a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). To derive the method, we first generate a high order perturbation of the original problem by using spline alternate relations. Then, we determine the approximate solution by forcing it to satisfy the resulting perturbed problem at the grid points of the spline. Convergence analysis of the method is established through matrix approach. Four nonlinear examples are considered to demonstrate the accuracy and robustness of the method. The proposed method provides O(h6) superconvergent approximation to the solution of the problem under consideration, where h is the step size. This method produces significantly more accurate results than the two newly developed numerical schemes using the same B-spline functions as used in the present method, namely UCS method and NCS method. Moreover, the computational time of present method is compared with that of NCS method.

Fourth-order numerical method for the Riesz space fractional diffusion equation with a nonlinear source term

‎This paper aims to propose a high-order and accurate numerical scheme for the solution of the nonlinear diffusion equation with Riesz space fractional derivative. To this end, we first discretize the Riesz fractional derivative with a fourth-order finite difference method, then we apply a boundary value method (BVM) of fourth-order for the time integration of the resulting system of ordinary differential equations. The proposed method has a fourth-order of accuracy in both space and time components and is unconditionally stable due to the favorable stability property of BVM. The numerical results are compared with analytical solutions and with those provided by other methods in the literature. Numerical experiments obtained from solving several problems including fractional Fisher and fractional parabolic-type sine-Gordon equations show that the proposed method is an efficient algorithm for solving such problems and can arrive at the high-precision.

A high-order numerical scheme for solving nonlinear time fractional reaction-diffusion equations with initial singularity

We propose a high-order numerical scheme for nonlinear time fractional reaction-diffusion equations with initial singularity, where L2-1σ scheme on graded mesh is used to approximate Caputo fractional derivative and Legendre spectral method is applied to discrete spatial variable. We give the priori estimate, existence and uniqueness of numerical solution. Then the unconditional stability and convergence are proved. The rate of convergence is O(M−min⁡{rα,2}+N−m), which is obtained without extra regularity assumption on the exact solution. Numerical results are given to confirm the sharpness of error analysis.

Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy

We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on each subinterval. The method is shown to be unconditionally stable, and for general nonlinear equations, the uniform sharp numerical order $3-\nu$ can be rigorously proven for sufficiently smooth solutions at all time steps. The proof provides a general guide for proving the sharp order for higher-order schemes in the nonlinear case. Some numerical examples are given to validate our theoretical results.

A higher-order accurate difference approximation of singularly perturbed reaction-diffusion problem using grid equidistribution

This paper proposes a higher-order numerical approximation scheme to solve singularly perturbed reaction–diffusion boundary value problems. The proposed scheme is a combination of a fourth-order numerical difference method and classical central difference method applied on a non-equidistant grid. The non-equidistant grid is generated by equi-distributing a non-negative monitor function. The theoretical and empirical error analysis demonstrate that the proposed scheme has fourth-order uniform convergence with respect to the perturbation parameter.

Time-domain simulations of sound propagation in a flow duct with extended-reacting liners

A general methodology to simulate acoustic propagation in ducts with extended-reacting liners in the time domain is presented, including a generic perforated sheet on the air-material interface. The Linearized Euler Equations (LEE) with a mean flow profile are solved in the duct and the linearized equations on an equivalent fluid are solved in the liner material. The auxiliary differential equation method (ADE) is used to prevent the computation of convolution integrals, and leads to a formulation compatible with high-order numerical schemes. The methodology is illustrated for the case of liners consisting of rigid-frame porous materials and a prototype of locally-resonant acoustic metamaterials. A one-dimensional (1D) test case is first used to validate the algorithm and assess the numerical error. The numerical order of the algorithm is the expected one, independently of the interface, as long as the number of poles retained in the partial fraction expansions involved in the formulation is high enough. The algorithm is then applied to a realistic two-dimensional (2D) configuration in a duct with flow, and is used to illustrate the restrained validity of the locally-reacting approximation. Finally, the impact of the flow Mach number on the acoustic performance of porous and metamaterial extended-reacting liners is briefly assessed.

An arbitrary high-order Spectral Difference method for the induction equation

We study in this paper three variants of the high-order Discontinuous Galerkin (DG) method with Runge-Kutta (RK) time integration for the induction equation, analysing their ability to preserve the divergence-free constraint of the magnetic field. To quantify divergence errors, we use a norm based on both a surface term, measuring global divergence errors, and a volume term, measuring local divergence errors. This leads us to design a new, arbitrary high-order numerical scheme for the induction equation in multiple space dimensions, based on a modification of the Spectral Difference (SD) method [1] with ADER time integration [2]. It appears as a natural extension of the Constrained Transport (CT) method. We show that it preserves ∇⋅B→=0 exactly by construction, both in a local and a global sense. We compare our new method to the 3 RKDG variants and show that the magnetic energy evolution and the solution maps of our new SD-ADER scheme are qualitatively similar to the RKDG variant with divergence cleaning, but without the need for an additional equation and an extra variable to control the divergence errors.

Model order reduction for efficient numerical room acoustic simulations with parametrized boundaries

This study presents a new model order reduction technique applied to room acoustic simulations using a high-order numerical scheme based on the spectral element method. The goal is to efficiently simulate iterative design processes in room acoustics, where the room acoustics with different boundary absorption properties are evaluated much quickly without solving the full wave equation. The wave-based methods are highly accurate but expensive when simulating high-frequencies and long simulation times in large and complex geometries, which are needed for industrial applications. Hence, it remains a key challenge to improve the computational performance of room acoustics simulations with different boundary material possibilities. With model order reduction, the boundary condition can be parametrized in the subsequent modelling. This allows to reduce the dimensionality of a wave-based simulator without compromising the transient wave propagation accuracy while enabling significant reductions of computational costs after initial training using proper orthogonal decomposition. We provide evidence that the combination of the high-order numerical scheme with model order reduction has significant potential value for building acoustics in terms of computational efficiency and accuracy.

Aerodynamic and aeroacoustic performance investigations on modified H-rotor Darrieus wind turbine

The present paper investigates the aerodynamic and aeroacoustic characteristics of the H-rotor Darrieus vertical axis wind turbine (VAWT) combined with very promising energy conversion and steering technology; a fixed guide-vanes. The main scope of the current work is to enhance the aerodynamic performance and assess the noise production accomplished with such enhancement. The studies are carried out in two phases; the first phase is a parametric 2D CFD simulation employing the unsteady Reynolds-averaged Navier-Stokes (URANS) approach to optimize the design parameters of the guide-vanes. The second phase is a 3D CFD simulation of the full turbine using a higher-order numerical scheme and a hybrid RANS/LES (DDES) method. The guide-vanes show a superior power augmentation, about 42% increase in the power coefficient at λ = 2.75, with a slightly noisy operation and completely change the signal directivity. A remarkable difference in power coefficient is observed between 2D and 3D models at the high-speed ratios stems from the 3D effect. As a result, a 3D simulation of the capped Darrieus turbine is carried out, and then a noise assessment of such configuration is assessed. The results show a 20% increase in power coefficient by using the cap, without significant change in the noise signal.

A new family of weighted one-parameter flux reconstruction schemes

The flux reconstruction (FR) approach offers a flexible framework for describing a range of high-order numerical schemes; including nodal discontinuous Galerkin and spectral difference schemes. This is accomplished through the use of so-called correction functions. In this study we employ a weighted Sobolev norm to define a new extended family of FR correction functions, the stability of which is affirmed through Fourier analysis. Several of the schemes within this family are found to exhibit reduced dissipation and dispersion overshoot. Moreover, many of the new schemes possess higher CFL limits whilst maintaining the expected rate of convergence. Numerical experiments with homogeneous linear convection and Burgers’ turbulence are undertaken, and the results observed to be in agreement with the theoretical findings.

Flux reconstruction using Jacobi correction functions in discontinuous spectral element method

A flux reconstruction approach is implemented in the spectral difference (SD) framework, which we refer to it as discontinuous spectral element method (DSEM), to improve the accuracy and stability of numerical simulations. A new class of correction functions is developed based on a weighted orthogonality condition, which leads to the introduction of Jacobi correction functions. Jacobi correction functions can construct the well-known DSEM and staggered-grid version of nodal discontinuous Galerkin spectral element method (DGSEM) as well as a broad range of other high-order numerical schemes with a variety of numerical characteristics, such as high numerical dissipation to suppress aliasing-driven errors, super accuracy, and solution boundedness in shock prediction. Three single-parameter families of Jacobi correction functions are recognized, and a von Neumann analysis is performed to acquire their wave propagation properties. The order of accuracy and stability criterion of a broad range of schemes are calculated, and the most super-accurate and stable numerical schemes are identified and later validated through a set of numerical experiments on a one-dimensional advection equation. The solution unboundedness issue of the high-order DSEM scheme is discussed through simulating the Burgers equation. An accuracy test performed on the Burgers equation revealed that the new staggered-grid flux reconstruction approach could acquire a higher accuracy than that of the energy stable flux reconstruction framework on a collocated grid. The two-dimensional extension of the new framework is also examined by simulation of a non-linear Euler system of equations for the isentropic vortex in free-stream flow. The staggered-grid flux reconstruction using the Jacobi correction function can precisely reconstruct the DSEM approach for non-linear two-dimensional hyperbolic equations for all polynomial orders.

Thermo-magneto-convection and mechanism of multiple eddy formation due to applied magnetic field in a lid-driven cavity

Magnetohydrodynamic study of eddy formation and heat transfer in a bottom heated two-dimensional lid-driven cavity with the insulated sidewall is reported in this paper. We consider the classic problem of a two-dimensional lid driven cavity in the presence of an applied external magnetic field. In the standard nonlinear Navier–Stokes equation and the energy equation, appropriate terms are added to model the applied magnetic field and thermo-magnetic convection. And we have derived a higher-order numerical scheme that can provide solutions that are fourth-order accurate. The scheme is implemented with suitable fourth-order accurate boundary conditions. The square LDC enclosure is filled with a fluid of Pr = 6.2. The top lid of the cavity is kept moving with constant velocity. The effect of magnetic field on heat transfer and eddy formation is discussed for 0 ≤ Ha ≤ 250, and 103 ≤ Gr ≤ 104 and the results are presented in terms of contour plots of streamlines, isotherms, Lorentz force, and the kinetic energy. The physical mechanism for forming multiple recirculation zones is understood and attributed to two opposite forces acting in the fluid about the center-line of the cavity, which acts like a couple (and hence rotational flow). This is discovered by computing the Lorentz force acting in the entire domain of fluid flow. Similarly, from the computation of kinetic energy of the fluid at all the grid points in the fluid region, it is revealed that conduction is the dominant heat transfer mechanism at higher magnetic fields. At the same time, convection is the prime mode at low magnetic fields. It is ascertained that with an increase in magnetic field intensity, velocity profiles are suppressed, and heat transfer stabilizes, giving rise to the formation of recirculation zones in the cavity.

Gauge invariant canonical symplectic algorithms for real-time lattice strong-field quantum electrodynamics

A class of high-order canonical symplectic structure-preserving geometric algorithms are developed for high-quality simulations of the quantized Dirac-Maxwell theory based strong-field quantum electrodynamics (SFQED) and relativistic quantum plasmas (RQP) phenomena. With minimal coupling, the Lagrangian density of an interacting bispinor-gauge fields theory is constructed in a conjugate real fields form. The canonical symplectic form and canonical equations of this field theory are obtained by the general Hamilton’s principle on cotangent bundle. Based on discrete exterior calculus, the gauge field components are discreted to form a cochain complex, and the bispinor components are naturally discreted on a staggered dual lattice as combinations of differential forms. With pull-back and push-forward gauge covariant derivatives, the discrete action is gauge invariant. A well-defined discrete canonical Poisson bracket generates a semi-discrete lattice canonical field theory (LCFT), which admits the canonical symplectic form, unitary property, gauge symmetry and discrete Poincaré subgroup, which are good approximations of the original continuous geometric structures. The Hamiltonian splitting method, Cayley transformation and symmetric composition technique are introduced to construct a class of high-order numerical schemes for the semi-discrete LCFT. These schemes involve two degenerate fermion flavors and are locally unconditional stable, which also preserve the geometric structures. Admitting Nielsen-Ninomiya theorem, the continuous chiral symmetry is partially broken on the lattice. As an extension, a pair of discrete chiral operators are introduced to reconstruct the lattice chirality. Equipped with statistically quantization-equivalent ensemble models of the Dirac vacuum and non-trivial plasma backgrounds, the schemes are expected to have excellent performance in secular simulations of relativistic quantum effects, where the numerical errors of conserved quantities are well bounded by very small values without coherent accumulation. The algorithms are verified in detail by numerical energy spectra. Real-time LCFT simulations are successfully implemented for the nonlinear Schwinger mechanism induced e-e+ pairs creation and vacuum Kerr effect, where the nonlinear and non-perturbative features captured by the solutions provide a complete strong-field physical picture in a very wide range, which open a new door toward high-quality simulations in SFQED and RQP fields.

A high order numerical method for solving Caputo nonlinear fractional ordinary differential equations

<abstract><p>In this paper, we construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations. Firstly, we use the piecewise Quadratic Lagrange interpolation method to construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations, and then analyze the local truncation error of the high order numerical scheme. Secondly, based on the local truncation error, the convergence order of $ 3-\theta $ order is obtained. And the convergence are strictly analyzed. Finally, the numerical simulation of the high order numerical scheme is carried out. Through the calculation of typical problems, the effectiveness of the numerical algorithm and the correctness of theoretical analysis are verified.</p></abstract>

A local discontinuous Galerkin method for nonlinear parabolic SPDEs

In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove theL2-stability of the numerical scheme for fully nonlinear equations. Optimal error estimates (O(h(k+1))) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degreekare used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method.

Numerical methods for stochastic differential equations based on Gaussian mixture

We develop in this work a numerical method for stochastic differential equations (SDEs) with weak second order accuracy based on Gaussian mixture. Unlike the conventional higher order schemes for SDEs based on It\^o-Taylor expansion and iterated It\^o integrals, the proposed scheme approximates the probability measure $\mu(X^{n+1}|X^n=x_n)$ by a mixture of Gaussians. The solution at next time step $X^{n+1}$ is then drawn from the Gaussian mixture with complexity linear in the dimension $d$. This provides a new general strategy to construct efficient high weak order numerical schemes for SDEs.

High order numerical schemes for transport equations on bounded domains

This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [10] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In additional, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.