Simple Finite Difference Scheme Research Articles

Output-feedback control of viscous liquid–tank system and its numerical approximation

We solve the output-feedback stabilization problem for a tank with a liquid modeled by the viscous Saint-Venant PDE system. The control input is the acceleration of the tank and a Control Lyapunov Functional methodology is used. The measurements are the tank position and the liquid level at the tank walls. The control scheme is a combination of a state feedback law with functional observers for the tank velocity and the liquid momentum. Four different types of output feedback stabilizers are proposed. A full-order observer and a reduced-order observer are used in order to estimate the tank velocity while the unmeasured liquid momentum is either estimated by using an appropriate scalar filter or is ignored. The reduced order observer differs from the full order observer because it omits the estimation of the measured tank position. Exponential convergence of the closed-loop system to the desired equilibrium point is achieved in each case. An algorithm is provided that guarantees that a robotic arm can move a glass of water to a pre-specified position no matter how full the glass is, without spilling water out of the glass, without residual end point sloshing and without measuring the water momentum and the glass velocity. Finally, the efficiency of the proposed output feedback laws is validated by numerical examples, obtained by using a simple finite-difference numerical scheme. The properties of the proposed, explicit, finite-difference scheme are determined.

Hybrid multi-grid parallelisation of WAVEWATCH III model on spherical multiple-cell grids

Spherical Multiple-Cell (SMC) grid is an unstructured grid, supporting flexible domain shapes and multi-resolutions. It retains the quadrilateral cells as in the latitude-longitude (lat-lon) grid so that simple finite-difference schemes can be used. Sub-timesteps are applied on refined cells and grid cells are merged at high latitudes to relax the CFL restriction. A fixed reference direction is used in polar regions to solve vector polar problems. The SMC grid was implemented in the WAVEWATCH III (WW3) wave model in 2012 as an alternative for the lat-lon grid and updated in the latest WW3 V6.07. The WW3 model is parallelised by wave spectral component decomposition (CD) in MPI mode, which has a limit on number of MPI ranks. Hybrid or combined MPI-OpenMP parallelisation may extend the node usage but the OpenMP scalability flattens out beyond a few threads. Another parallelisation method that combines CD with domain decomposition (DD) is enabled in WW3 model by a multi-grid framework for further extension of node usage. Those regular lat-lon grid parallelisation options are gradually added to the SMC grid and this article reports the recent extension of the SMC grid into the multi-grid framework with hybrid parallelisation. The flexible domain shape of the SMC grid allows optimised domain splitting and minimised boundary exchanges. The combined CD-DD method is tested on SMC sub-grids with various hybrid node-thread combinations. Results indicate that switching from MPI to hybrid MPI-OpenMP mode can halve the global model elapsed time and using hybrid CD-DD on 3 SMC sub-grids may reduce it further by 30%. Elapsed time for one model day run is reduced from about 3 min on 12 nodes to less than 1 min on 90 or 180 nodes. Besides, the hybrid multi-grid method reduces memory demand on one computing node and allows future model updates for higher resolutions.

KdV and BBM models in open cylindrical channel under precession

The Korteweg–de Vries partial differential equation that has nonlinear-dispersion balance was derived under potential conditions to match the case of a single Kelvin mode that was noticed revolving on the outer periphery of an open cylindrical channel under precession conditions, which is assumed the solitary wave case in the channel. This led to a new version of the equation with a forcing term that includes the tilt effect with coefficients include the rotation effect. It was solved numerically using Fourier transformation methods for space discretization and the fourth order Runge–Kutta method for time discretization; the results were in a good match with the experiment. The rotational case led to a new Benjamin–Bona–Mahony equation that has variable coefficients with time and space mainly coming from the Coriolis effect in the axial direction of motion, with a forcing term comes from the gravity force. It was also solved numerically using a simple implicit finite difference scheme. This equation has two versions, one in terms of the velocity and one in terms of the amplitude. The first was compared with the bore velocity signal, which reflected the cnoidal type of waves, and the results were in a satisfactory match with the extracted signals; the second one was tracked with time to see the role Coriolis and gravity forces play on the single Kelvin wave form.

KdV Equation Model in Open Cylindrical Channel under Precession

A new model for Korteweg and de-Vries equation (KdV) is derived. The system under study is an open channel consisting of two concentric cylinders, rotating about their vertical axis, which is tilted by slope tau from the inertial vertical z, in uniform rate {Omega }_{1}=tau Omega, and the whole tank is elevated over other table rotating at rate Omega. Under these conditions, a set of Kelvin waves is formed on the free surface depending on the angle of tilt, characterized by the slope tau, volume of water, and rotation rate. The resonant mode in the system appears in the form of a single Kelvin solitary wave, whose amplitude satisfies the Korteweg-de Vries equation with forced term. The equation was derived following classical perturbation methods, the additional term made the equation a non-integrable one, that cannot be solved without the help of numerical methods. Invoking the simple finite difference scheme method, it was found that the numerical results are in a good agreement with the experiment.

Numerical Algorithms for Elastoplacity: Finite Elements Code Development and Implementation of the Mohr–Coulomb Law

Two numerical algorithms for solving elastoplastic problems with the finite element method are presented. The first deals with the implementation of the return mapping algorithm and is based on a fixed-point algorithm. This method rewrites the system of elastoplasticity non-linear equations in a form adapted to the fixed-point method. The second algorithm relates to the computation of the elastoplastic consistent tangent matrix using a simple finite difference scheme. A first validation is performed on a nonlinear bar problem. The results obtained show that both numerical algorithms are very efficient and yield the exact solution. The proposed algorithms are applied to a two-dimensional rockfill dam loaded in plane strain. The elastoplastic tangent matrix is calculated by using the finite difference scheme for Mohr–Coulomb’s constitutive law. The results obtained with the developed algorithms are very close to those obtained via the commercial software PLAXIS. It should be noted that the algorithm’s code, developed under the Matlab environment, offers the possibility of modeling the construction phases (i.e., building layer by layer) by activating the different layers according to the imposed loading. This algorithmic and implementation framework allows to easily integrate other laws of nonlinear behaviors, including the Hardening Soil Model.

Non-standard Finite Difference Based Numerical Method for Viscous Burgers’ Equation

Simple and accurate non-standard finite difference (NSFD) schemes are proposed for solving the viscous Burgers’ equation with and without Cole–Hopf transformation. The stability and positivity of the schemes are discussed in detail, and the schemes are very user friendly in implementation. For some classic examples for the viscous Burgers’ equation, simulation outcomes are validated through comparison with exact solutions for small and large values of kinematic viscosity. Numerical results from the NSFD schemes and corresponding standard finite difference methods are tabulated and compared with results from some existing methods. The proposed schemes give relatively accurate results, with comparatively less observed absolute errors.

Pressure - velocity projection method with mixed type approximation for Oseen discrete operator

We are considering the problem of divergence - free projection in rectangular domains for the Oseen (stationary Navier-Stokes) operator. It uses nodal Finite Element method to reconstruct pressure and Finite Difference method for scalar functions and vector functions for other parts of the equations. This allows us to use simple finite difference schemes for linear part and WENO-type schemes for the the nonlinear part in the space of discrete velocities. The boundary conditions are formulated in Finite Volume conservative approach for velocity and in Finite Element approach for pressure. We illustrate that the convergence rate depends on the smoothness of the projected function and that the projection method allows one to obtain divergence-free solutions with desired tolerance. We numerically analize the projection operator that acts in the space of divergence operator image. It is shown that all eigenvalues of the operator are less than unity for Oseledets-type wall boundary conditions for velocity and zero Neumann conditions for the Poisson equation. We show that for smooth functions we obtain forth order convergence in divergence. Finally we demonstrate the method by solving the stationary lid driven cavity problem for the 2D Oseen operator.

PDE acceleration: a convergence rate analysis and applications to obstacle problems

This paper provides a rigorous convergence rate and complexity analysis for a recently introduced framework, called PDE acceleration, for solving problems in the calculus of variations and explores applications to obstacle problems. PDE acceleration grew out of a variational interpretation of momentum methods, such as Nesterov’s accelerated gradient method and Polyak’s heavy ball method, that views acceleration methods as equations of motion for a generalized Lagrangian action. Its application to convex variational problems yields equations of motion in the form of a damped nonlinear wave equation rather than nonlinear diffusion arising from gradient descent. These accelerated PDEs can be efficiently solved with simple explicit finite difference schemes where acceleration is realized by an improvement in the CFL condition from $$\mathrm{d}t\sim \mathrm{d}x^2$$ for diffusion equations to $$\mathrm{d}t\sim \mathrm{d}x$$ for wave equations. In this paper, we prove a linear convergence rate for PDE acceleration for strongly convex problems, provide a complexity analysis of the discrete scheme, and show how to optimally select the damping parameter for linear problems. We then apply PDE acceleration to solve minimal surface obstacle problems, including double obstacles with forcing, and stochastic homogenization problems with obstacles, obtaining state-of-the-art computational results.

Analysis and computation of an optimality equation arising in an impulse control problem with discrete and costly observations

The framework of discrete and costly observations provides an attractive mathematical tool for stochastic control of biological and ecological systems in natural environment. However, this framework is still germinating, and its applications in these research areas are still rare. In this framework, the system dynamics to be controlled is continuous-time, while the observations by the decision-maker are discrete-time. The control problem of the system dynamics has a time-hybridized type nature, by which the optimality equation has a different form from the conventional ones. The objectives as well as contributions of this paper are formulation of the optimality equation focusing on population management problems, its detailed mathematical analysis, numerical discretization, and a demonstrative application. We show that the optimality equation is solvable uniquely and its solution can be approximated recursively. A numerical method to discretize the optimality equation is established with a simple finite difference scheme and the recursive formula. Mathematical analysis results such as monotonicity, stability, and convergence of the numerical method are discussed as well. The present model is then applied to a population management problem. Finally, an advanced problem subject to model ambiguity is also analyzed.

Optimal harvesting policy of an inland fishery resource under incomplete information

AbstractA new mathematical model for finding the optimal harvesting policy of an inland fishery resource under incomplete information is proposed in this paper. The model is based on a stochastic control formalism in a regime‐switching environment. The incompleteness of information is due to uncertainties involved in the body growth rate of the fishery resource: a key biological parameter. Finding the most cost‐effective harvesting policy of the fishery resource ultimately reduces to solving a terminal and boundary value problem of a Hamilton‐Jacobi‐Bellman equation: a nonlinear and degenerate parabolic partial differential equation. A simple finite difference scheme for solving the equation is then presented, which turns out to be convergent and generates numerical solutions that comply with certain theoretical upper and lower bounds. The model is finally applied to the management of Plecoglossus altivelis, a major inland fishery resource in Japan. The regime switching in this case is due to the temporal dynamics of benthic algae, the main food of the fish. Model parameter values are identified from field measurement results in 2017. Our computational results clearly show the dependence of the optimal harvesting policy on the river environmental and biological conditions. The proposed model would serve as a mathematical tool for fishery resource management under uncertainties.

Hybrid Active Contours Driven by Edge and Region Fitting Energies Based on p-Laplace Equation

By using local and global image information, a novel active contour method based on the p-Laplace equation for image segmentation is proposed in this paper. The force term in the evolution equation incorporates global, local and edge information of the image. The global and local terms drag the contour towards object boundaries precisely and take care of the contour movement when it is away from the object. Meanwhile, the integration of edge stopping function in this force term ensures the stoppage of contour at object boundaries to avoid boundary leakage problem. The variable exponent p-Laplace energy is used for smoothness of the level set to detect the exact object boundaries in the presence of complex topological changes and deep depression. Finally, the adaptive force and p-Laplace energy term are jointly integrated into a level set by using a simple finite difference scheme to build the final evolution equation for the method. The proposed method has strong capability to accurately segment the images having noise, intensity inhomogeneity, and complex object boundaries. Moreover, the proposed method overcomes the problem arise from contour initialization as the evolution of contour is independent of the initialization of level set function. Experimental results on synthetic, real and medical images along with two publicly available databases validate the robustness and effectiveness of the proposed method.

A novel variational model for noise robust document image binarization

Abstract Binarization is a fundamental problem in document image analysis systems. Different from current thresholding techniques, a novel variational model is proposed for noise robust document image binarization in this work, which is inspired by two classical variational models that have been successfully applied in image segmentation and denoising. In our variational model, the energy functional consists of three terms: data fidelity term, binary classification term and regularization term. As a result, the proposed model is capable of performing image binarization and suppressing noise simultaneously. Concretely, we firstly design a variational model for the original document image, the minimizer of which is our desired binarization result. Secondly, the gradient descent flow equation of our model is derived by means of variational principle. Lastly, a simple finite difference scheme, time forward and space center difference, is employed to solve the gradient descent flow equation. Extensive experiments are conducted on three types of document image datasets to validate our model qualitatively and quantitatively. The experiment results not only demonstrate the effectiveness of our approach, but also verify its noise and illumination robustness.

A local radial basis function collocation method for band structure computation of phononic crystals with scatterers of arbitrary geometry

• A new numerical algorithm based on the LRBFCM is proposed. • The stability of the LRBFCM is greatly increased. • The boundary or interface condition of complex geometry can be easily treated. • The improved LRBFCM is applied to the PCs with a scatterer of complex geometry. A numerical algorithm based on the local radial basis function collocation method (LRBFCM) is developed to efficiently compute the derivatives of primary field quantities. Instead of a direct calculation of the derivatives by partial differentiation of the shape functions as in traditional numerical approaches, the derivative calculation in the present work is performed using a simple finite difference scheme with an introduced fictitious node. The developed algorithm is geometrically very flexible and can be easily applied to the continuity and boundary conditions of arbitrary geometries, which require an accurate derivative computation of the primary field quantities. The developed LRBFCM are applied to phononic crystals with scatterers of arbitrary geometry, which has not yet been reported before to the authors’ knowledge. A few examples for anti-plane elastic wave propagation are modelled to validate the developed LRBFCM. A comparison with finite element modelling shows that the present method is efficient and flexible.

A simple finite-difference scheme for handling topography with the first-order wave equation

One approach to incorporate topography in seismic finite-difference codes is a local modification of the difference operators near the free surface. An earlier paper described an approach for modelling irregular boundaries in a constant-density acoustic finite-difference code, based on the second-order formulation of the wave equation that only involves the pressure. Here, a similar method is considered for the first-order formulation in terms of pressure and particle velocity, using a staggered finite-difference discretization both in space and in time. In one space dimension, the boundary conditions consist in imposing antisymmetry for the pressure and symmetry for particle velocity components. For the pressure, this means that the solution values as well as all even derivatives up to a certain order are zero on the boundary. For the particle velocity, all odd derivatives are zero. In 2D, the 1-D assumption is used along each coordinate direction, with antisymmetry for the pressure along the coordinate and symmetry for the particle velocity component parallel to that coordinate direction. Since the symmetry or antisymmetry should hold along the direction normal to the boundary rather than along the coordinate directions, this generates an additional numerical error on top of the time stepping errors and the errors due to the interior spatial discretization. Numerical experiments in 2D and 3D nevertheless produce acceptable results

A simple finite-difference scheme for handling topography with the second-order wave equation

ABSTRACTThe presence of topography poses a challenge for seismic modeling with finite-difference codes. The representation of topography by means of an air layer or vacuum often leads to a substantial loss of numerical accuracy. A suitable modification of the finite-difference weights near the free surface can decrease that error. An existing approach requires extrapolation of interior solution values to the exterior while using the boundary condition at the free surface. However, schemes of this type occasionally become unstable and may be impossible to implement with highly irregular topography. One-dimensional extrapolation along coordinate lines results in a simple and efficient scheme. The stability of the 1D scheme is improved by ignoring the interior point nearest to the boundary during extrapolation in case its distance to the boundary is less than half a grid spacing. The generalization of the 1D scheme to more than one dimension requires a modification if the boundary intersects the finite-difference stencil on both sides of the central evaluation point and if there are not enough interior points to build the finite-difference stencil. Examples for the 2D constant-density acoustic case with a fourth-order finite-difference scheme demonstrate the method’s capability. Because the 1D assumption is not valid in two dimensions if the boundary does not follow grid lines, the formal numerical accuracy is not always obtained, but the method can handle highly irregular topography.

A few simple CFD models for flow simulation around groyne

The flow processes around a groyne structure is a complex and challenging topic in computational hydraulics and has brought attention of many researchers in the last few decades. In comparison to experimental studies, numerical modeling can provide more details about the flow characteristics at less cost. In the present work, performances of four simple finite difference schemes, i.e., Lax diffusive scheme, predictor corrector scheme, modified predictor corrector scheme and Beam and Warming scheme are compared in simulation of flow around groyne structure. The numerical results are validated with available experimental data and the best model is found out in terms of accuracy and efficiency.

Temperature analysis of partially heated steel members in fire

Analytical methods for the evaluation of temperature distribution of steel members in fire are readily available in codes of practice. These methods are usually based on uniform sections under fully engulfed fire conditions. In reality, structural members are often subject to partial heating fire conditions such as members passing through multiple fire compartments or subject to localised fires. At present, little guidance is given for temperature calculation of steel members subject to partial heating fire conditions. Evaluating the temperature distribution of steel members in such situations is important as the results may determine the extent of fire protection required for these members. This paper describes a numerical study on the temperature distribution of a partially heated steel member using a simple finite difference scheme with parametrically coded generic elements. The results are compared with the approximated method given in Eurocodes whereby the average temperature of a partially heated steel member can be calculated. It is shown that the proposed finite difference scheme can accurately predict the temperature distribution of steel members and provides a means for assessing their fire protection requirements.

Active contours for image segmentation using complex domain‐based approach

A complex domain-based approach of active contour model has been proposed for image segmentation which deforms iteratively to partition an image into various useful regions. A new region-based complex pressure force function has been designed which gives shape to the dynamic active contour using image forces, and efficiently control the propagation of moving interface. This model makes the level set function binary, uses Gaussian smoothing kernel to regulate and avoid re-initialisation procedure. The working scheme of the model is as: the real image data is converted into complex data by iota (i) times and the average iota (i) times of horizontal and vertical components of gradient is inserted in the model to develop complete complex gradient. The proposed model can be implemented by simple finite difference scheme. The efficiency and robustness of the authors model have been verified.

A new compact finite difference scheme for solving the complex Ginzburg–Landau equation

The complex Ginzburg–Landau equation is often encountered in physics and engineering applications, such as nonlinear transmission lines, solitons, and superconductivity. However, it remains a challenge to develop simple, stable and accurate finite difference schemes for solving the equation because of the nonlinear term. Most of the existing schemes are obtained based on the Crank–Nicolson method, which is fully implicit and must be solved iteratively for each time step. In this article, we present a fourth-order accurate iterative scheme, which leads to a tri-diagonal linear system in 1D cases. We prove that the present scheme is unconditionally stable. The scheme is then extended to 2D cases. Numerical errors and convergence rates of the solutions are tested by several examples.

MPI-AMRVAC FOR SOLAR AND ASTROPHYSICS

In this paper we present an update on the open source MPI-AMRVAC simulation toolkit where we focus on solar- and non-relativistic astrophysical magneto-fluid dynamics. We highlight recent developments in terms of physics modules such as hydrodynamics with dust coupling and the conservative implementation of Hall magnetohydrodynamics. A simple conservative high-order finite difference scheme that works in combination with all available physics modules is introduced and demonstrated at the example of monotonicity preserving fifth order reconstruction. Strong stability preserving high order Runge-Kutta time steppers are used to obtain stable evolutions in multidimensional applications realizing up to fourth order accuracy in space and time. With the new distinction between active and passive grid cells, MPI-AMRVAC is ideally suited to simulate evolutions where parts of the solution are controlled analytically, or have a tendency to progress into or out of a stationary state. Typical test problems and representative applications are discussed, with an outlook to follow-up research. Finally, we discuss the parallel scaling of the code and demonstrate excellent weak scaling up to 30 000 processors allowing to exploit modern petascale infrastructure.