scholarly journals On second-order accuracy schemes for modeling of plasma oscillations

2020 ◽  
pp. 115-128
Author(s):  
Е.В. Чижонков
Keyword(s):  
Numerical Experiments ◽  
Second Order ◽  
Power Laser ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Plasma Oscillations ◽  
First Order ◽  
High Power Laser Pulse ◽  
Quantitative Characteristics ◽  
Free Plasma

Для моделирования колебаний холодной плазмы как в нерелятивистском случае, так и с учетом релятивизма предложены модификации классических разностных схем второго порядка точности: метода МакКормака и двухэтапного метода Лакса-Вендроффа. Ранее для подобных расчетов в эйлеровых переменных была известна только схема первого порядка точности. Для задачи о свободных плазменных колебаниях, инициированных коротким мощным лазерным импульсом, с целью тестирования представленных схем проведены численные эксперименты по сохранению энергии и других величин. Сделан вывод о достоверности численного анализа колебаний как на основе схемы МакКормака, так и на основе схемы Лакса-Вендроффа, однако для расчетов долгоживущих процессов первая схема более предпочтительна. Теоретическое исследование аппроксимации и устойчивости вместе с экспериментальным наблюдением за количественными характеристиками погрешности для наиболее чувствительных величин существенно повышает достоверность вычислений. Ключевые слова: численное моделирование, плазменные колебания, эффект опрокидывания, схемы МакКормака и Лакса-Вендроффа, порядок точности разностной схемы, законы сохранения. For modeling cold plasma oscillations in the non-relativistic and relativistic cases, some modifications of classical difference schemes of the second order of accuracy are proposed: the McCormack method and the two-stage Lax-Wendroff method. Previously, only the first-order accuracy scheme was known for calculations in Euler variables. For the problem of free plasma oscillations initiated by a short high-power laser pulse, the results of numerical experiments on energy conservation and other quantities were performed in order to test the proposed schemes. It is concluded that the numerical analysis of oscillations is reliable both for the McCormack scheme and for the Lax-Wendroff scheme however, for the calculation of long-lived processes, the first scheme is more preferable. The theoretical analysis of approximation and stability together with experimental observations of quantitative characteristics of errors for the most sensitive quantities significantly increases the reliability of calculations.

scholarly journals Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

2020 ◽  
Vol 25 (6) ◽  
pp. 997-1014
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Difference Scheme ◽  
Numerical Method ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Finite Difference Schemes ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
First Order ◽  
The Difference ◽  
Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

scholarly journals High Order Godunov Type Multimesh Method for 3d Impact Problems of Elastoplastic Media

2021 ◽  
Keyword(s):  
Three Dimensional ◽  
Second Order ◽  
The Body ◽  
Godunov Scheme ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Sharp Interface Method ◽  
Impact Problems ◽  
Elastoplastic Media ◽  
Compact Stencil

A numerical method for calculating the three-dimensional processes of impact interaction of elastoplastic bodies under large displacements and deformations based on the multi mesh sharp interface method and modified Godunov scheme is presented. To integrate the equations of dynamics of an elastoplastic medium, the principle of splitting in space and in physical processes is used. The solutions of the Riemann problem for first and second order accuracy for compact stencil for an elastic medium in the case of an arbitrary stress state are obtained and presented, which are used at the “predictor” step of the Godunov scheme. A modification of the scheme is described that allows one to obtain solutions in smoothness domains with a second order of accuracy on a compact stencil for moving Eulerian-Lagrangian grids. Modification is performed by converging the areas of influence of the differential and difference problems for the Riemann’s solver. The “corrector” step remains unchanged for both the first and second order accuracy schemes. Three types of difference grids are used. The first – a moving surface grid – consists of a continuous set of triangles that limit and accompany the movement of bodies; the size and number of triangles in the process of deformation and movement of the body can change. The second – a regular fixed Eulerian grid – is limited to a surface grid; separately built for each body; integration of equations takes place on this grid; the number of cells in this grid can change as the body moves. The third grid is a set of local Eulerian-Lagrangian grids attached to each moving triangle of the surface from the side of the bodies and allowing obtain the parameters on the boundary and contact surfaces. The values of the underdetermined parameters in cell’s centers near the contact boundaries on all types of grids are interpolated. Comparison of the obtained solutions with the known solutions by the Eulerian-Lagrangian and Lagrangian methods, as well as with experimental data, shows the efficiency and sufficient accuracy of the presented three-dimensional methodology.

CTCS Schemes for Second Order Wave Equation: Numerical Results and Spectral Analysis

2021 ◽  
Vol 55 ◽  
pp. 47-65
Author(s):  
Appanah Rao Appadu ◽  
Gysbert Nicolaas de Waal
Keyword(s):  
Spectral Analysis ◽  
Wave Equation ◽  
Numerical Experiments ◽  
Finite Difference Methods ◽  
Second Order ◽  
Numerical Results ◽  
Order Accuracy ◽  
Constant Coefficients ◽  
Second Order Wave Equation ◽  
The One

IIn this paper, two finite difference methods are used to solve the one-dimensional second order wave equation with constant coefficients subject to specified initial and boundary conditions. Two numerical experiments are considered. The two methods are Central in Time and Central in Space scheme with second order accuracy in both time and space, abbreviated as CTCS (2,2) and Central in Time and Central in Space scheme with second order accuracy in time and fourth order accuracy in space, abbreviated as CTCS (2,4). Properties such as consistency and stability are studied. We also perform spectral analysis of dispersive and dissipative properties of the two methods. Two numerical experiments are considered, and the numerical results are displayed.

scholarly journals Study on Accuracy of the High-Resolution Schemes

2014 ◽  
Vol 6 ◽  
pp. 905053
Author(s):  
Yawen Tang ◽  
Bo Yu ◽  
Jianyu Xie ◽  
Jingfa Li ◽  
Peng Wang
Keyword(s):  
High Resolution ◽  
Numerical Solution ◽  
Richardson Extrapolation ◽  
Second Order ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Numerical Examples ◽  
High Resolution Schemes ◽  
Artificial Diffusion ◽  
Convection Dominated

The high-resolution (HR) schemes have been widely used as they can achieve the numerical solution without oscillation and artificial diffusion, especially for convection-dominated problems. However, there still have arguments about the order of accuracy of HR schemes, especially about the extreme value of the solution. In this paper, it is proved that any HR scheme designed in the NVD diagram has second-order accuracy when its combined segments totally locate in the BAIR region. In other words, it has been verified in our study that the segments, which have low-order accuracy when independently employed, have at least second-order accuracy when locate in BAIR region by analysis of two implementation methods of HR scheme and also a number of numerical examples. Meanwhile Richardson extrapolation has been used to estimate the order of accuracy of HR schemes which achieve the same conclusion.

The Special Form of Second Order Accuracy N-S Equations for Incompressible Fluid

2014 ◽  
Vol 644-650 ◽  
pp. 1644-1647
Author(s):  
Zhan Song Li ◽  
Shi Jiang Zhu
Keyword(s):  
Laminar Flow ◽  
Turbulent Flow ◽  
Incompressible Fluid ◽  
Special Form ◽  
Higher Order ◽  
Second Order ◽  
Time And Space ◽  
Order Accuracy ◽  
First Order

Classic N-S equation has first order accuracy in both of time and space. It has only the terms of first order, without the terms of second or higher order. These terms are relative in time and space steps. The time and space steps, as basic elements of fluid research, should be only some finite quantities and not be infinitely near to zero as defined in mathematics. If the terms of second or higher order can be ignored depends on the value of the corresponding derivative multiplied. Compared with terms of first order, the terms of second or higher order can be ignored under the condition of laminar flow. However, under the condition of turbulent flow, these can’t be ignored yet. When turbulent flow develops fully, the terms of first order, compared with terms of second order, can be ignored. So, it is why classic N-S equations aren’t closed when they are used to analyze turbulent flow. On the basic, many different special forms of the second order accuracy N-S equations of incompressible fluid are derived.

Novel Linear Implicit Integration Method and its Application to Linear/Nonlinear Structural System

2014 ◽  
Vol 1065-1069 ◽  
pp. 1535-1539
Author(s):  
Chuan Guo Jia ◽  
Yan Xing Liu ◽  
Ying Min Li ◽  
Min Mao Liao
Keyword(s):  
Computational Efficiency ◽  
Unconditional Stability ◽  
Second Order ◽  
Order Accuracy ◽  
Dynamic Simulations ◽  
Order Of Accuracy ◽  
Engineering Research ◽  
Time Integrators ◽  
Shear Type ◽  
The Stability

Dynamic simulations of structures to determine their seismic performance is an essential part of civil engineering research. Time integrators of increasing sophistication has been elaborated over the last few decades to achieve higher order accuracy, unconditional stability, computational efficiency and high-frequency dissipation. This paper tries to extend 1-stage Rosenbrock-based integrator to an integrator of second-order accuracy without losing computational efficiency and unconditional stability. Initially, 1-stage Rosenbrock integrator is introduced and its order of accuracy is studied theoretically. According to accuracy analysis, a force term at the end of current step is considered, resulting a novel integrator of second-order accuracy. Moreover, the stability of the proposed method is studied by means of the energy method. To investigate its performance for nonlinear structures, numerical simulations are conducted on a shear-type structure including a pendulum.

scholarly journals A Computational Framework for A First-Order System of Conservation Laws in Thermoelasticity

2021 ◽  
Author(s):  
◽  
Ataollah Ghavamian
Keyword(s):  
Finite Element ◽  
Conservation Laws ◽  
Total Energy ◽  
Second Order ◽  
Factor H ◽  
Irreversible Processes ◽  
Order Accuracy ◽  
Computational Framework ◽  
First Order ◽  
Displacement Based

It is evidently not trivial to analytically solve practical engineering problems due to their inherent (geometrical and/or material) nonlinearities. Moreover, experimental testing can be extremely costly, time-consuming and even dangerous, in some cases. In the past few decades, therefore, numerical techniques have been progressively developed and utilised in order to investigate complex engineering applications through computer simulations, in a cost-effective manner.An important feature of a numerical methodology is how to approximate a physical domain into a computational domain and that, typically, can be carried out via mesh-based and particle-based approximations, either of which manifest with a different range of capabilities. Due to the geometrical complexity of many industrial applications (e.g. biomechanics, shape casting, metal forming, additive manufactur-ing, crash simulations), a growing attraction has been received by tetrahedral mesh generation, thanks to Delaunay and advancing front techniques [1, 2]. Alternatively, particle-based methods can be used as they offer the possibility of tackling specific applications in which mesh-based techniques may not be efficient (e.g. hyper velocity impact, astrophysics, failure simulations, blast).In the context of fast thermo-elastodynamics, modern commercial packages are typically developed on the basis of second order displacement-based finite element formulations and, unfortunately, that introduces a series of numerical shortcomings such as reduced order of convergence for strains and stresses in comparison with displacements and the possible onset of numerical instabilities (e.g. detrimental locking, hour-glass modes, spurious pressure oscillations).To rectify these drawbacks, a mixed-based set of first order hyperbolic conservation laws for isothermal elastodynamics was presented in [3–6], in terms of the linear momentum p per unit undeformed volume and the minors of the deformation, namely, the deformation gradient F , its co-factor H and its Jacobian J. Taking inspiration of these works [4, 7] and in order to account for irreversible processes, the balance of total energy (also known as the first law of thermodynamics) is incorporated to the set of physical laws used to describe the deformation process. This, in general, can be expressed in terms of the entropy density η or total energy density E by which the Total Lagrangian entropy-based and total energy-based formulations {p, F , H, J, η or E} are established, respectively. Interestingly, taking advantage of the conservation formulation framework, it is possible to bridge the gap between solid dynamics and Computational Fluid Dynamics (CFD) by exploiting available CFD techniques in the context of solid dynamics.From a computational standpoint, two distinct and extremely competitive spatial discretisations are employed, namely, mesh-based Vertex-Centred Finite Volume Method (VCFVM) and meshless Smooth Particle Hydrodynamics (SPH). A linear reconstruction procedure together with a slope limiter is employed in order to ensure second order accuracy in space whilst avoiding numerical oscillations in the vicinity of sharp gradients, respectively. Crucially, the discontinuous solution for the conservation variables across (dual) control volume interfaces or between any pair of particles is approximated via an acoustic Riemann solver. In addition, a tailor-made artificial compressibility algorithm and an angular momentum preservation scheme are also incorporated in order to assess same limiting scenarios.The semi-discrete system of equations is then temporally discretised using a one-step two-stage Total Variation Diminishing (TVD) Runge-Kutta time integrator, providing second order accuracy in time. The geometry is also monolithically updated to be only used for post-processing purposes.Finally, a wide spectrum of challenging examples is presented in order to assess both the performance and applicability of the proposed schemes. The new formulation is proven to be very efficient in nearly incompressible thermo-elasticity in comparison with classical finite element displacement-based approaches. The proposed computational framework provides a good balance between accuracy and speed of computation.

scholarly journals On approximation of two-dimensional potential and singular operators

2019 ◽  
Vol 2019 (1) ◽  
pp. 68-83
Author(s):  
Charyyar Ashyralyyev ◽  
Sedanur Efe
Keyword(s):  
Numerical Calculation ◽  
Singular Integral ◽  
Numerical Experiments ◽  
Spline Approximation ◽  
Integral Operators ◽  
Quadrature Formulas ◽  
Two Dimensional ◽  
Order Of Accuracy ◽  
First Order ◽  
Order Spline

Abstract The purpose of this paper is the construction of second-order of accuracy quadrature formulas for the numerical calculation of the Vekua types two-dimensional potential and singular integral operators in the unit disk of complex plane. We propose quadrature formulas for these integrals which based on first-order spline approximation of two-dimensional function. MATLAB programs are used for numerical experiments in test examples.

A SECOND-ORDER ADAPTIVE ARBITRARY LAGRANGIAN–EULERIAN METHOD FOR THE COMPRESSIBLE EULER EQUATIONS

2009 ◽  
Vol 23 (04) ◽  
pp. 583-601
Author(s):  
Y. J. WANG ◽  
N. ZHAO ◽  
C. W. WANG ◽  
D. H. WANG
Keyword(s):  
Numerical Experiments ◽  
Adaptive Mesh ◽  
Time Discretization ◽  
Second Order ◽  
Arbitrary Lagrangian Eulerian ◽  
Finite Volume Schemes ◽  
Order Accuracy ◽  
Specific Internal Energy ◽  
Ale Methods ◽  
Lagrangian Scheme

Most of finite volume schemes in the Arbitrary Lagrangian–Eulerian (ALE) method are constructed on the staggered mesh, where the momentum is defined at the nodes and the other variables (density, pressure and specific internal energy) are cell-centered. However, this kind of schemes must use a cell-centered remapping algorithm twice which is very inefficient. Furthermore, there is inconsistent treatment of the kinetic and internal energies.1 Recently, a new class of cell-centered Lagrangian scheme for two-dimensional compressible flow problems has been proposed in Ref. 2. The main new feature of the algorithm is the introduction of four pressures on each edge, two for each node on each side of the edge. This scheme is only first-order accurate. In this paper, a second-order cell-centered conservative ENO Lagrangian scheme is constructed by using an ENO-type approach to extend the spatial second-order accuracy. Time discretization is based on a second-order Runge–Kutta scheme. Combining a conservative interpolation (remapping) method3,4 with the second-order Lagrangian scheme, a kind of cell-centered second-order ALE methods can be obtained. Some numerical experiments are made with this method. All results show that our method is effective and have second-order accuracy. At last, in order to further increase the resolution of shock regions, we use an adaptive mesh generation based on the variational principle5 as a rezoned strategy for developing a class of adaptive ALE methods. Numerical experiments are also presented to valid the performance of the proposed method.

scholarly journals NUMERICAL SOLVING UNSTEADY SPACE-FRACTIONAL PROBLEMS WITH THE SQUARE ROOT OF AN ELLIPTIC OPERATOR

2016 ◽  
Vol 21 (2) ◽  
pp. 220-238 ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Elliptic Operator ◽  
Finite Element Approximation ◽  
Second Order ◽  
Evolutionary Equation ◽  
Element Approximation ◽  
Square Root ◽  
Dimensional Problem ◽  
Order Accuracy ◽  
First Order ◽  
Fractional Equation

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two- level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adams scheme. More general problems for the equation with convective terms are considered, too. The results of numerical experiments are presented for a model two-dimensional problem.

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