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.

Numerical solution of a source identification problem: Almost coercivity

2019 ◽  
Vol 27 (4) ◽  
pp. 457-468 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Abdullah Said Erdogan ◽  
Ali Ugur Sazaklioglu
Keyword(s):  
Numerical Solution ◽  
Source Identification ◽  
Identification Problem ◽  
Second Order ◽  
Numerical Results ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Accuracy Difference ◽  
Coercive Stability

Abstract The present paper is devoted to the investigation of a source identification problem that describes the flow in capillaries in the case when an unknown pressure acts on the system. First and second order of accuracy difference schemes are presented for the numerical solution of this problem. Almost coercive stability estimates for these difference schemes are established. Additionally, some numerical results are provided by testing the proposed methods on an example.

scholarly journals Numerical Modeling of Self-Consistent Dynamics of Shallow and Ground Waters

2021 ◽  
pp. 45-62
Author(s):  
Sergey Khrapov
Keyword(s):  
Numerical Modeling ◽  
Numerical Solution ◽  
Software Package ◽  
Simulation Software ◽  
Second Order ◽  
Order Of Accuracy ◽  
Ground Waters ◽  
Spatially Inhomogeneous ◽  
Tvd Method ◽  
Joint Dynamics

A mathematical and numerical model of the joint dynamics of shallow and ground waters has been built, which takes into account the nonlinear dynamics of a liquid, water absorption from the surface into the ground, filtration currents in the ground, and water seepage from the ground back to the surface. The dynamics of shallow waters is described by the Saint-Venant equations, taking into account the spatially inhomogeneous distributions of the terrain, the coefficients of bottom friction and infiltration, as well as non-stationary sources and flows of water. For the numerical integration of Saint-Venant’s equations, the well-tested CSPH-TVD method of the second order of accuracy is used, the parallel CUDA algorithm of which is implemented as a software package “EcoGIS-Simulation” for high-performance computing on supercomputers with graphic coprocessors (GPU). The dynamics of groundwater is described by the nonlinear Bussensk equation, generalized to the case of a spatially inhomogeneous distribution of the parameters of the porous medium and the surface of the aquiclude (the boundary between water-permeable and low-permeable soils). The numerical solution of this equation is built on the basis of a finite-difference scheme of the second order of accuracy, the CUDA algorithm of which is integrated into the calculation module of the “EcoGIS-Simulation” software package and is consistent with the main stages of the CSPH-TVD method. The relative deviation of the numerical solution from the exact solution of the nonlinear Boussinesq equation does not exceed 10−4–10−5. The paper compares the results of numerical modeling of the dynamics of groundwater with analytical solutions of the linearized Bussensk equation used as calculation formulas in the methods for predicting the level of groundwater in the vicinity of water bodies. It is shown that the error of these methods is several percent even for the simplest case of a plane-parallel flow of groundwater with a constant backwater. Based on the results obtained, it was concluded that the proposed method for numerical modeling of the joint dynamics of surface and ground waters can be more versatile and efficient (it has significantly better accuracy and productivity) in comparison with the existing methods for calculating flooding zones, especially for hydrodynamic flows with complex geometry and nonlinear interaction of counter fluid flows arising during seasonal floods during flooding of vast land areas.

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.

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 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 ANALYSIS OF UPWIND AND HIGH‐RESOLUTION SCHEMES FOR SOLVING CONVECTION DOMINATED PROBLEMS IN POROUS MEDIA

2006 ◽  
Vol 11 (4) ◽  
pp. 451-474 ◽  
Author(s):  
V. Starikovičius ◽  
R. Čiegis
Keyword(s):  
Porous Media ◽  
High Resolution ◽  
Hyperbolic Equations ◽  
Test Problems ◽  
Differential Approximation ◽  
Upwind Schemes ◽  
High Resolution Schemes ◽  
Differential Approximation Method ◽  
Parallel Arrays ◽  
Convection Dominated

The conservation laws governing the multiphase flows in porous media are often convection‐dominated and have a steep fronts that require accurate resolution. Standard discretization methods of the convection terms do not perform well for such problems. The main aim of this work is to analyze the use of upwind and high‐ resolution schemes in such cases. First, we use a first differential approximation method to perform a theoretical analysis of a standard upwind approximation and different time stepping schemes for the linear hyperbolic equations in 1‐ and 2D. Next, we present a popular approach to reduce the amount of numerical diffusion introduced by upwind approximation ‐ high‐resolution schemes. We compare our implementation of one of the recently proposed central‐upwind schemes against the upwind schemes on several test problems based on Buckley‐Leverett equation and discuss the results. Finally, a parallel version of central‐upwind scheme in 2D is presented. It was implemented using our C++ library of parallel arrays ‐ ParSol.

scholarly journals Generalized finite difference solution for the Motz Problem

2020 ◽  
Vol 36 (4) ◽  
Author(s):  
C. Chávez ◽  
F. Domíngez ◽  
S. Lucas-Martínez ◽  
J. Tinoco-Martínez ◽  
D. Santana
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Optimality Condition ◽  
Finite Difference Scheme ◽  
Second Order ◽  
Order Accuracy ◽  
Numerical Examples ◽  
Finite Difference Solution ◽  
General Difference ◽  
Difference Solution

In this paper, it is presented a formulation of a generalized finite difference scheme to solve the Motz problem. It is based on a general difference scheme defined by an optimality condition, which has been developed to solve Poisson-like equations whose domains are approximated by a wide variety of grids over general regions. Numerical examples showing second-order accuracy of the calculated solutions are presented.

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