Linearly Implicit Approximations of Diffusive Relaxation Systems

We present a fast first order explicit time integration scheme for solving parabolic problems in mechanics via standard numerical methods in space using irregular grids, such as unstructured finite element meshes, or grids containing elements or cells of very different sizes. The new scheme extends one of the explicit FIC-Time (EFT) integration methods derived by the authors in [23] that allow considerable larger time steps than the forward-Euler (FE) scheme. The new EFT scheme overcomes the limitations in the time step size of explicit time integration schemes for irregular grids containing large and small elements. A variable time step is used for eliminating the oscillations near Dirichlet boundaries when large time steps are used. The advantages of the new EFT scheme versus the FE scheme are shown in one-, two- and three-dimensional transient heat conduction problems using irregular finite element grids.

The numerical simulation of nonlinear dispersive waves is a central research topic of many investigations in the nonlinear wave community. Simple and robust solvers are needed for numerical studies of water waves as well. The main difficulties arise in the numerical approximation of high order derivatives and in severe stability restrictions on the time step, when explicit schemes are used. In this study we propose new relaxed system formulations which approximate the initial dispersive wave equation. However, the resulting relaxed system involves first order derivatives only and it is written in the form of an evolution problem. Thus, many standard methods can be applied to solve the relaxed problem numerically. In this article we illustrate the application of the new relaxed scheme on the classical Korteweg-de Vries equation as a prototype of stiff dispersive PDEs.

This paper develops a numerical procedure to accelerate the convergence of the Favre-averaged non-linear harmonic (FNLH) method. The scheme provides a unified mathematical framework for solving the sparse linear systems formed by the mean flow and the time-linearized harmonic flows of FNLH in an explicit or implicit fashion. The approach explores the similarity of the sparse linear systems of FNLH and leads to a memory-efficient procedure, so that its memory consumption does not depend on the number of harmonics to compute. The proposed method has been implemented in the industrial computational fluid dynamics solver Hydra. Three test cases are used to conduct a comparative study of explicit and implicit schemes in terms of convergence, computational efficiency, and memory consumption. Comparisons show that the implicit scheme yields better convergence than the explicit scheme and is also roughly 7–10 times more computationally efficient than the explicit scheme with four levels of multigrid. Furthermore, the implicit scheme consumes only approximately 50% of the memory required by the explicit scheme with four levels of multigrid. Compared with the full-annulus unsteady Reynolds-averaged Navier–Stokes simulations, the implicit scheme produces comparable results to URANS with computational time and memory consumption that are two orders of magnitude smaller.

The heat conduction equation is a combination of first-order and second-order differential equations. Solving first-order differential equations is necessary to examine temperature as a function of time. Meanwhile, solving second-order differential equations is needed to examine temperature as a function of space. The heat flux equation is based on Fourier's law, which shows that temperature is a function of time and space. Understanding heat conduction can be improved by building a heat propagation model on the Solar ERK dryer plate. Analysis of heat propagation on the drying plate used the Finite Difference Approach (FDA) method with explicit and implicit schemes. With an explicit scheme, the FDA method calculates the temperature (T) at a point on the spatial derivative term, when T is at time t, while the implicit scheme calculates T at a point on the space derivative term when T is at time t+Δt. Heat propagation at each time change was analyzed by developing a program using the MATLAB 17 application. The results of the analysis show that there are differences in heat propagation between the explicit and implicit schemes. The convergence and stability of calculations in explicit schemes are unstable, causing problems at the time step. Meanwhile, the implicit scheme is carried out simultaneously on all nodes so that convergence and stability are easily maintained, and there are no time-step limitations.

Nonlinear diffusion filtering in image processing is usually performed with explicit schemes. They are only stable for very small time steps, which leads to poor efficiency and limits their practical use. Based on a discrete nonlinear diffusion scale-space framework we present semi-implicit schemes which are stable for all time steps. These novel schemes use an additive operator splitting (AOS), which guarantees equal treatment of all coordinate axes. They can be implemented easily in arbitrary dimensions, have good rotational invariance and reveal a computational complexity and memory requirement which is linear in the number of pixels. Examples demonstrate that, under typical accuracy requirements, AOS schemes are at least ten times more efficient than the widely used explicit schemes.

A fully explicit marching-on-in-time scheme for solving the time domain magnetic field integral equation is proposed. The unknown current density induced on the surface of the scatterer is expanded using a higher-order Nystrom method in space and Lagrange interpolation in time. The resulting system is cast in the form of an ordinary differential equation and integrated in time using a predictor-corrector scheme to obtain the unknown expansion coefficients. Numerical results demonstrate that the proposed explicit scheme can use the same time step size as its implicit counterpart without sacrificing stability and is five times faster under low-frequency excitation (i.e., for large time step).

Summary form only given. Many materials in the nature or fabricated in laboratories exhibit nonlinear behavior where the permittivity and/or permeability is a function of the magnitude of the electric or magnetic field intensity. Such nonlinearity results in a nonlinear system of Maxwell's equations, which are more challenging to solve. Generally speaking, there are two kinds of methods commonly adopted to solve nonlinear problems. The first kind of methods solves the linearized versions of the original problems, while the second kind of methods deals with the nonlinear problems directly. Although more difficult to solve with a high efficiency, the second kind of methods is more desirable in order to capture the nonlinear phenomenon more accurately. Among various nonlinear solvers developed in the applied mathematics community, the Newton-Raphson method is widely used due to its second-order convergence, if started from an initial guess sufficiently close to the true solution. However, in a real application, there is no straightforward way to choose a good initial guess. As a result, the Newton-Raphson method could be unstable or divergent in solving complicated problems. Another difficulty of applying the Newton-Raphson method is that the Jacobian matrix needs to be updated and solved in every Newton iteration, which could be very time consuming for solving large problems. When the nonlinear electromagnetic problems are formulated and solved with the finite element method in time domain (J. M. Jin, The Finite Element Method in Electromagnetics, Second Edition. New York: Wiley, 2002), the numerical solution at a previous time step can be used as a natural choice for the initial guess of the Newton iteration at the successive time step. The convergent of the Newton-Raphson method can be achieved if the time step is small enough such that the solutions of two successive time steps are sufficiently close. In this presentation, we employ the time-domain finite element method (TDFEM) combined with the Newton-Raphson method and its variations to solve nonlinear electromagnetic problems. We first study the TDFEM formulation of nonlinear problems, and then show how it can be solved with Newton(-like) methods. We demonstrate and compare the performance of different nonlinear solvers including the basic Newton-Raphson method and its variations through a number of numerical examples.

A simplified finite volume lattice Boltzmann method for simulations of fluid flows from laminar to turbulent regime, Part II: Extension towards turbulent flow simulation

The explicit finite-difference scheme is widely used in seismic wave simulation. Using large time steps can significantly reduce the number of iterations and thus improve computational efficiency, but this approach faces two main challenges: reduced accuracy caused by temporal dispersion and restrictions imposed by the Courant-Friedrichs-Lewy (CFL) stability condition. Based on the wavefield iteration equation in matrix form, an explicit finite-difference method with time step n-tupling and an extended CFL limit is developed for acoustic wave simulation. By combining n successive iteration operators into a single-step operator, two types of n-tupling algorithms are constructed, effectively expanding the CFL stability limit by a factor of n and enabling time steps well beyond conventional thresholds. Both theoretical analysis and numerical experiments demonstrate that simulations with time step n-tupling achieve accuracy equivalent to conventional single-step methods while reducing computation time to approximately 1/n, resulting in a corresponding n-fold increase in overall computational efficiency. Time dispersion is suppressed using two complementary strategies: for relatively small base time steps, n-tupling inherently attains the accuracy of smaller steps; for relatively large base steps, a time-dispersion transform is applied to eliminate errors and maintain high numerical accuracy throughout the simulation.

An efficient and provable sequential quadratic programming method for American and swing option pricing

We present two novel classes of fully discrete energy-preserving algorithms for the sine-Gordon equation subject to Neumann boundary 条件.The cosine pseudo-spectral method is firstly used to develop structure-preserving spatial discretizations under two different meshes, which result in two finite-dimensional Hamiltonian ODE (ordinary differential equation) systems. Then we combine the prediction-correction Crank-Nicolson 格式 with the projection approach to arrive at fully discrete energy-preserving methods. Alternatively, we introduce a S variable to transform the initial model into a relaxation system, which allowsus to develop structure-preserving algorithms more easily. We then discretize the relaxation system directly by using the cosine pseudo-spectral method in space and the prediction-correction Crank-Nicolson 格式 in time to derive a new class of energy-preserving schemes. The proposed methods can be solved efficiently by the discrete cosine transform. Some benchmark examples and numerical comparisons are presented to demonstrate the accuracy, efficiency and superiority of the proposed schemes.

Partially explicit splitting scheme with explicit–implicit-null method for nonlinear multiscale flow problems

The YBJ equation (Young & Ben Jelloul, J. Marine Res., vol. 55, 1997, pp. 735–766) provides a phase-averaged description of the propagation of near-inertial waves (NIWs) through a geostrophic flow. YBJ is obtained via an asymptotic expansion based on the limit $\mathit{Bu}\rightarrow 0$, where $\mathit{Bu}$ is the Burger number of the NIWs. Here we develop an improved version, the YBJ+ equation. In common with an earlier improvement proposed by Thomas, Smith & Bühler (J. Fluid Mech., vol. 817, 2017, pp. 406–438), YBJ+ has a dispersion relation that is second-order accurate in $\mathit{Bu}$. (YBJ is first-order accurate.) Thus both improvements have the same formal justification. But the dispersion relation of YBJ+ is a Padé approximant to the exact dispersion relation and with $\mathit{Bu}$ of order unity this is significantly more accurate than the power-series approximation of Thomas et al. (2017). Moreover, in the limit of high horizontal wavenumber $k\rightarrow \infty$, the wave frequency of YBJ+ asymptotes to twice the inertial frequency $2f$. This enables solution of YBJ+ with explicit time-stepping schemes using a time step determined by stable integration of oscillations with frequency $2f$. Other phase-averaged equations have dispersion relations with frequency increasing as $k^{2}$ (YBJ) or $k^{4}$ (Thomas et al. 2017): in these cases stable integration with an explicit scheme becomes impractical with increasing horizontal resolution. The YBJ+ equation is tested by comparing its numerical solutions with those of the Boussinesq and YBJ equations. In virtually all cases, YBJ+ is more accurate than YBJ. The error, however, does not go rapidly to zero as the Burger number characterizing the initial condition is reduced: advection and refraction by geostrophic eddies reduces in the initial length scale of NIWs so that $\mathit{Bu}$ increases with time. This increase, if unchecked, would destroy the approximation. We show, however, that dispersion limits the damage by confining most of the wave energy to low $\mathit{Bu}$. In other words, advection and refraction by geostrophic flows does not result in a strong transfer of initially near-inertial energy out of the near-inertial frequency band.

In this paper, a multiscale approach with partially explicit time discretization is proposed. The idea is to use a partially explicit time scheme, considering a filtration problem in a fractured medium, where the implicit scheme is used for nodes whose subdomains contain fractures, and the explicit scheme is used for all others. In this way, it is possible to use a time step that is independent of the diffusion coefficient for fractures. Numerical results demonstrating high accuracy of calculations are presented.

A robust implicit–explicit acoustic-transport splitting scheme for two-phase flows