Standard Explicit Schemes Research Articles

Accelerating Transient Aerothermal Simulations of the BOLT II Flight Experiment

Aircraft with short flight durations may not reach thermal steady state due to limited time for aeroheating to conduct throughout their internal structure. Consequently, predictions of the solid temperature field require time-accurate simulation of the transient heat conduction. In this work, the use of an explicit super-time-stepping scheme to accelerate the time-accurate solution of the heat equation has been explored. The experience with this technique is documented here because, to the authors’ knowledge, it is not widely used in the hypersonics simulation community. The performance of the super-time-stepping scheme was investigated by comparison to a standard explicit scheme and an implicit time-differencing scheme. Both the super-time-stepping scheme and the implicit time-differencing scheme demonstrate significant speedup over the standard explicit scheme, with comparable wall-clock performance. The solid solver has been coupled with an existing open-source fluid solver to perform conjugate heat transfer simulations. The efficacy of the developed solver in simulating hypersonic aerothermal problems is demonstrated through verification and validation studies. By applying the coupled solver to the aerothermal analysis of the BOLT II flight experiment, the effectiveness of the super-time-stepping scheme in mitigating the computational expense associated with transient conjugate heat transfer simulations is demonstrated.

Acceleration of particle-in-cell simulations using sparse grid algorithms. I. Application to dual frequency capacitive discharges

The use of sparse particle-in-cell (PIC) algorithms to accelerate the standard explicit PIC scheme has recently been successfully applied in the context of single-frequency capacitively coupled plasma discharges [Garrigues et al., J. Appl. Phys. 129, 153303 (2021)]. We have extended the sparse PIC scheme to model dual-frequency capacitive discharges. Comparisons between standard and sparse PIC algorithms show that the plasma properties as well as the electron and ion distribution functions can be retrieved with a maximum error of 2%. This work opens the interest of using the sparse PIC algorithm to perform 2D and 3D simulations under real operating conditions of capacitively coupled plasma discharges.

Effects of confinement-induced non-Newtonian lubrication forces on the rheology of a dense suspension

In this work, we propose a functionalised bi-viscous lubrication model to study the material properties of concentrated non-Brownian suspensions and explore the possible confinement-induced non-Newtonian effects of the lubricant in the rheological response of this type of suspensions. From tribological studies, it is well-known that even macroscopically Newtonian liquids under strong confinement might exhibit properties which deviate significantly from their bulk behaviour. When two surfaces separated by an extremely small gap (still large compared to the molecular size) are sheared, strong shear-thinning of the lubricant viscosity at low shear-rates is observed, in spite of its Newtonian-like bulk response. This is connected to a significant increase of the zero-shear-rate viscosity under extreme confinement. We start from an effective lubrication algorithm recently proposed and develop a new gap-size-dependent interparticle bi-viscous lubrication model, able to capture qualitatively the main phenomenology of confined lubricants. We solve the lubrication interaction between particles iteratively via a semi-implicit splitting scheme. Since the handling of lubrication is made implicitly here, the method copes efficiently with large increases of the inter-particle effective viscosities, which would otherwise lead to simulation blow-up or the use of vanishing time-steps in standard explicit schemes. We analyse the rheological response of the suspension systematically in terms of model parameters. In contrast to pure Newtonian lubrication interactions, distinct shear-thinning and shear-thickening regimes in the relative suspension viscosity are observed, which are discussed in terms of particle microstructure coupled with the complex rheology of the confined lubricant. In addition, normal-stress response is negative in both N1 and N2, which is difficult to achieve with standard contact frictional models.

Strong convergence of adaptive time-stepping schemes for the stochastic Allen–Cahn equation

Abstract It is known from Beccari et al. (2019) that the standard explicit Euler-type scheme (such as the exponential Euler and the linear-implicit Euler schemes) with a uniform timestep, though computationally efficient, may diverge for the stochastic Allen–Cahn equation. To overcome the divergence, this paper proposes and analyzes adaptive time-stepping schemes, which adapt the timestep at each iteration to control numerical solutions from instability. The a priori estimates in $\mathscr{C}(\mathscr{O})$-norm and $\dot{H}^{\beta }(\mathscr{O})$-norm of numerical solutions are established provided the adaptive timestep function is suitably bounded, which plays a key role in the convergence analysis. We show that the adaptive time-stepping schemes converge strongly with order $\frac{\beta }{2}$ in time and $\frac{\beta }{d}$ in space with $d$ ($d=1,2,3$) being the dimension and $\beta \in (0,2]$. Numerical experiments show that the adaptive time-stepping schemes are simple to implement and at a lower computational cost than a scheme with the uniform timestep.

High order asymptotic preserving scheme for diffusive scaled linear kinetic equations with general initial conditions

Diffusive scaled linear kinetic equations appear in various applications, and they contain a small parameter ɛ that forces a severe time step restriction for standard explicit schemes. Asymptotic preserving (AP) schemes are those schemes that attain asymptotic consistency and uniform stability for all values of ɛ, with the time step restriction being independent of ɛ. In this work, we develop high order AP scheme for such diffusive scaled kinetic equations with both well-prepared and non-well-prepared initial conditions by employing IMEX-RK time integrators such as CK-ARS and A types. This framework is also extended to a different collision model involving advection-diffusion asymptotics, and the AP property is proved formally. A further extension of our framework to inflow boundaries has been made, and the AP property is verified. The temporal and spatial orders of accuracy of our framework are numerically validated in different regimes of ɛ, for all the models. The qualitative results for diffusion asymptotics, and equilibrium and non-equilibrium inflow boundaries are also presented.

An implicit–explicit time discretization for elastic wave propagation problems in plates

AbstractWe propose a new implicit–explicit scheme to address the challenge of modeling wave propagation within thin structures using the time‐domain finite element method. Compared to standard explicit schemes, our approach renders a time marching algorithm with a time step independent of the plate thickness and its associated discretization parameters (mesh step and order of approximation). Relying on the standard three dimensional elastodynamics equations, our strategy can be applied to any type of material, either isotropic or anisotropic, with or without discontinuities in the thickness direction. Upon the assumption of an extruded mesh of the plate‐like geometry, we show that the linear system to be solved at each time step is partially lumped thus efficiently treated. We provide numerical evidence of an adequate convergence behavior, similar to a reference solution obtained using the well‐known leapfrog scheme. Further numerical investigations show significant speed up factors compared to the same reference scheme, proving the efficiency of our approach for the configurations of interest.

Well-posed evolution of field theories with anisotropic scaling: the Lifshitz scalar field in a black hole space-time

Partial differential equations exhibiting an anisotropic scaling between space and time — such as those of Hořava-Lifshitz gravity — have a dispersive nature. They contain higher-order spatial derivatives, but remain second order in time. This is inconvenient for performing long-time numerical evolutions, as standard explicit schemes fail to maintain convergence unless the time step is chosen to be very small. In this work, we develop an implicit evolution scheme that does not suffer from this drawback, and which is stable and second-order accurate. As a proof of concept, we study the numerical evolution of a Lifshitz scalar field on top of a spherically symmetric black hole space-time. We explore the evolution of a static pulse and an (approximately) ingoing wave-packet for different strengths of the Lorentz-breaking terms, accounting also for the effect of the angular momentum eigenvalue and the resulting effective centrifugal barrier. Our results indicate that the dispersive terms produce a cascade of modes that accumulate in the region in between the Killing and universal horizons, indicating a possible instability of the latter.

A uniformly accurate numerical method for a class of dissipative systems

We consider a class of ordinary differential equations mixing slow and fast variations with varying stiffness (from non-stiff to strongly dissipative). Such models appear for instance in population dynamics or propagation phenomena. We develop a multi-scale approach by splitting the equations into a micro part and a macro part, from which the original stiffness has been removed. We then show that both parts can be simulated numerically with uniform order of accuracy using standard explicit numerical schemes. As a result, solving the problem in its micro-macro formulation can be done with a cost and an accuracy independent of the stiffness. This work is also a preliminary step towards the application of such methods to hyperbolic partial differential equations and we will indeed demonstrate that our approach can be successfully applied to two discretized hyperbolic systems (with and without non-linearities), though with some ad-hoc regularization.

Stable IMEX schemes for a Nitsche-based approximation of elastodynamic contact problems. Selective mass scaling interpretation

We introduce some IMEX schemes (implicit-explicit schemes with an implicit term being linear) for approximating elastodynamic contact problems when the contact condition is taken into account with a Nitsche method. We develop a theoretical and numerical study of the properties of the schemes, especially in terms of stability, provide some numerical comparisons with standard explicit and implicit scheme and propose some improvements to obtain a more reliable approximation of motion for large time steps. We also show how selective mass scaling techniques can be interpreted as IMEX schemes.

An asymptotic preserving well-balanced scheme for the isothermal fluid equations in low-temperature plasmas at low-pressure

We present a novel numerical scheme for the efficient and accurate solution of the isothermal two-fluid (electron + ion) equations coupled to Poisson's equation for low-temperature plasmas at low-pressure. The model considers electrons and ions as separate fluids, comprising the electron inertia and space charge regions. The discretization of this system with standard explicit schemes is constrained by very restrictive time steps and cell sizes related to the resolution of the Debye length, electron plasma frequency, and electron sound waves. Both sheath and electron inertia are fundamental to fully explain the physics in low-pressure and low-temperature plasmas. However, most of the phenomena of interest for fluid models occur at speeds much slower than the electron thermal speed and are quasi-neutral, except in small charged regions. A numerical method that is able to simulate efficiently and accurately all these regimes is a challenge due to the multiscale character of the problem. In this work, we present a scheme based on the Lagrange-projection operator splitting that preserves the asymptotic regime where the plasma is quasi-neutral with massless electrons. As a result, the quasi-neutral regime is treated without the need of an implicit solver nor the resolution of the Debye length and electron plasma frequency. Additionally, the scheme proves to accurately represent the dynamics of the electrons both at low speeds and when the electron speed is comparable to the thermal speed. In addition, a well-balanced treatment of the ion source terms is proposed in order to tackle problems where the ion temperature is very low compared to the electron temperature. The scheme significantly improves the accuracy both in the quasi-neutral limit and in the presence of plasma sheaths when the Debye length is resolved. In order to assess the performance of the scheme in low-temperature plasmas conditions, we propose two specifically designed test-cases: a quasi-neutral two-stream periodic perturbation with analytical solution and a low-temperature discharge that includes sheaths. The numerical strategy, its accuracy, and computational efficiency are assessed on these two discriminating configurations.

Dispersive Contour-Path MNL-FDTD Scheme for Fast Analysis of Waveguide Metamaterials

A semi-implicit dispersive thin-wire finite-difference time-domain (FDTD) scheme is proposed for fast analysis of waveguide metamaterials. The proposed scheme is based on combining the Newmark-beta method with Makinen’s improved thin-wire FDTD model based on the contour-path integral of Maxwell’s equations around a wire corrected with scaling factors and Wait’s surface-impedance boundary condition for thin wires of finite conductivity. Its stability condition is determined by only the two meshes in the transverse plane to the wire axis. This feature allows using larger mesh steps than a wire radius without time step reducing. In order to verify the scheme, we demonstrate four device applications: plasmonic parallel-plate waveguide filled with only positive dielectrics, edge Fabry–Perot resonator, waveguide metatronic high-pass filter, and coaxial-to-waveguide matching with an epsilon-near-zero narrow channel. Its stability, accuracy, and computational efficiency are assessed in comparison with the standard FDTD explicit scheme, analytical solutions, and existing experimental data.

Convergence and qualitative properties of modified explicit schemes for BSDEs with polynomial growth

The theory of Forward–Backward Stochastic Differential Equations (FBSDEs) paves a way to probabilistic numerical methods for nonlinear parabolic PDEs. The majority of the results on the numerical methods for FBSDEs relies on the global Lipschitz assumption, which is not satisfied for a number of important cases such as the Fisher–KPP or the FitzHugh–Nagumo equations. Furthermore, it has been shown in [Ann. Appl. Probab. 25 (2015) 2563–2625] that for BSDEs with monotone drivers having polynomial growth in the primary variable $y$, only the (sufficiently) implicit schemes converge. But these require an additional computational effort compared to explicit schemes. This article develops a general framework that allows the analysis, in a systematic fashion, of the integrability properties, convergence and qualitative properties (e.g., comparison theorem) for whole families of modified explicit schemes. The framework yields the convergence of some modified explicit scheme with the same rate as implicit schemes and with the computational cost of the standard explicit scheme. To illustrate our theory, we present several classes of easily implementable modified explicit schemes that can computationally outperform the implicit one and preserve the qualitative properties of the solution to the BSDE. These classes fit into our developed framework and are tested in computational experiments.

An implicit gas-kinetic scheme for turbulent flow on unstructured hybrid mesh

In this study, an implicit scheme for the gas-kinetic scheme (GKS) on the unstructured hybrid mesh is proposed. The Spalart–Allmaras (SA) one equation turbulence model is incorporated into the implicit gas-kinetic scheme (IGKS) to predict the effects of turbulence. The implicit macroscopic governing equations are constructed and solved by the matrix-free lower-upper symmetric-Gauss–Seidel (LU-SGS) method. To reduce the number of cells and computational cost, the hybrid mesh is applied. A modified non-manifold hybrid mesh data(NHMD) is used for both unstructured hybrid mesh and uniform grid. Numerical investigations are performed on different 2D laminar and turbulent flows. The convergence property and the computational efficiency of the present IGKS method are investigated. Much better performance is obtained compared with the standard explicit gas-kinetic scheme. Also, our numerical results are found to be in good agreement with experiment data and other numerical solutions, demonstrating the good applicability and high efficiency of the present IGKS for the simulations of laminar and turbulent flows.

Two-level schemes for the advection equation

The advection equation is the basis for mathematical models of continuum mechanics. In the approximate solution of nonstationary problems it is necessary to inherit main properties of the conservatism and monotonicity of the solution. In this paper, the advection equation is written in the symmetric form, where the advection operator is the half-sum of advection operators in conservative (divergent) and non-conservative (characteristic) forms. The advection operator is skew-symmetric. Standard finite element approximations in space are used. The standard explicit two-level scheme for the advection equation is absolutely unstable. New conditionally stable regularized schemes are constructed, on the basis of the general theory of stability (well-posedness) of operator-difference schemes, the stability conditions of the explicit Lax–Wendroff scheme are established. Unconditionally stable and conservative schemes are implicit schemes of the second (Crank–Nicolson scheme) and fourth order. The conditionally stable implicit Lax–Wendroff scheme is constructed. The accuracy of the investigated explicit and implicit two-level schemes for an approximate solution of the advection equation is illustrated by the numerical results of a model two-dimensional problem.

MPC-based reference governor control of a continuous stirred-tank reactor

Optimal control of a CSTR represents a challenging task. The proposed paper discusses two issues. The first one addresses control of pH in a chemical vessel, where the reaction between sodium hydroxide and acetic acid occurs. The objective here is to improve control performance of a well tuned PI controller. It will be shown that this can be achieved by introducing a reference governor scheme. The second problem, elaborated in this paper, is the implementation of the reference governor paradigm. Concretely, we aim to design a fast and cheap MPC-based feedback controller. To achieve these goals, we exploit the region-less explicit technique, which efficiently reduces memory footprint issues of standard explicit MPC schemes. Such MPC-based reference governor was employed to control pH in the chemical vessel. Its control performance is compared with conventional PI controller. Finally, comparison of implementation requirements of region-less and region-based explicit techniques is investigated.

An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes

We present a new mixed explicit implicit time stepping scheme for solving the linear advection equation on a Cartesian cut cell mesh. We use a standard second-order explicit scheme on Cartesian cells away from the embedded boundary. On cut cells, we use an implicit scheme for stability. This approach overcomes the small cell problem--that standard schemes are not stable on the arbitrarily small cut cells--while keeping the cost fairly low. We examine several approaches for coupling the schemes in one dimension. For one of them, which we refer to as flux bounding, we can show a TVD result for using a first-order implicit scheme. We also describe a mixed scheme using a second-order implicit scheme and combine both mixed schemes by an FCT approach to retain monotonicity. In the second part of this paper, extensions of the second-order mixed scheme to two and three dimensions are discussed and the corresponding numerical results are presented. These indicate that this mixed scheme is second-order accurate in $$L^1$$L1 and between first- and second-order accurate along the embedded boundary in two and three dimensions.

Comparative analysis of different numerical schemes in solute trapping simulations by using the phase-field model with finite interface dissipation

Two different numerical schemes, the standard explicit scheme and the time-elimination relaxation one, in the framework of phase-field model with finite interface dissipation were employed to investigate the solute trapping effect in a Si-4.5 at.% As alloy during rapid solidification. With the equivalent input, a unique solute distribution under the steady state can be obtained by using the two schemes without restriction to numerical length scale and interface velocity. By adjusting interface width and interface permeability, the experimental solute segregation coefficients can be well reproduced. The comparative analysis of advantages and disadvantages in the two numerical schemes indicates that the time-elimination relaxation scheme is preferable in one-dimensional phase-field simulation, while the standard explicit scheme seems to be the only choice for two- or three dimensional phase-field simulation. Furthermore, the kinetic phase diagrams in the Si-As system were predicted by using the phase-field simulation with the time-elimination relaxation scheme.

The numerical analysis of the weighted average method for heat conduction equation in one dimension

In this paper, we will consider the weighted average method for heat conduction equation in one dimension. When θ � 1 the truncation error of the scheme is O(τ + h 2 ) ;i fθ = 1 the truncation error of the scheme is O(τ 2 + h 2 ) .W hen 0 θ 1 , the scheme is stable if and only ifγ 1 (1 − 2θ) �1 and when 1 θ 1, the scheme is stable for all γ .T he numerical experiments are illustrated at the last. u ∂x2 , ( a> 0), −∞ <x< +∞, 0 t T u(x, 0) = ϕ(x), −∞ <x< +∞ (1) For this model problem an explicit difference method is very straightforward in use, and the analysis of its error is easily accomplished by the use of a maximum principle, or by Fourier analysis (5). Standard explicit schemes for parabolic equations are not very convenient for computing practice due to the fact that they have strong restrictions on a time step. More promising explicit schemes are as- sociated with explicit-implicit splitting of the problem operator. These schemes belong to the class of uncondition ally stable schemes, but they demonstrate bad approximation properties. In (8) P.N. Vabi shchevich proposes a multilevel modification of the alternating triangle method, which demonstrates better property -es in terms of accuracy. In (6) Rooholah Abedian proposes a new WENO finite differ- enceprocedurefor nonlineardegenerateparabolicequationswhich may containdiscontinuoussolutions. Their scheme is based on the method of lines, with a high-order accurate conservative approximation to each of the diffuse -on terms. In (7) Allaberen Ashyralyev considers the r-modified Crank-Nicholson difference schemes for the approximate solution of the ultra-parabolic equations. Explicit scheme is simple but conditionally stable. Implicit methods are stable, however they can take much longer to compute than explicit methods. So we consider to built the weighted average scheme to combine the advantages of both type scheme. In the present paper, we are going to analyze the truncation error and stability of the weighted average scheme in theory or in numeration. The paper is organized as following: The numerical schemes for heat conduction equations in one dimension; numerical experiments; summary.

An improved Milstein method for stiff stochastic differential equations

To solve the stiff stochastic differential equations, we propose an improved Milstein method, which is constructed by adding an error correction term to the Milstein scheme. The correction term is derived from an approximation of the difference between the exact solution of stochastic differential equations and the Milstein continuous-time extension. The scheme is proved to be strongly convergent with order one and is as easy to implement as standard explicit schemes but much more efficient for solving stiff stochastic problems. The efficiency and the advantage of the method lie in its very large stability region. For a linear scalar test equation, it is shown that the mean-square stability domain of the method is much bigger than that of the Milstein method. Finally, numerical examples are reported to highlight the accuracy and effectiveness of the method.

Boundary integral method for the flow of vesicles with viscosity contrast in three dimensions

We propose numerical algorithms for the simulation of the dynamics of three-dimensional vesicles suspended in viscous Stokesian fluid. Our method is an extension of our previous work (S.K. Veerapaneni et al., 2011) [37] to flows with viscosity contrast. This generalization requires a change in the boundary integral formulation of the solution, in which a double-layer Stokes integral is introduced, and leads to changes in the fluid dynamics due to the viscosity contrast of the vesicles, which can no longer be efficiently resolved with existing algorithms.In this paper we describe the algorithms needed to handle flows with viscosity contrast accurately and efficiently. We show that a globally semi-implicit method does not have any time-step stability constraint for flows with single and multiple vesicles with moderate viscosity contrast and the computational cost per simulation unit time is comparable to or less than that of an explicit scheme. Automatic oversampling adaptation enables us to achieve high accuracy with very low spectral resolution. We conduct numerical experiments to investigate the stability, accuracy, and the computational cost of the algorithms. Overall, our method achieves several orders of magnitude speed-up compared to the standard explicit schemes.