scholarly journals Convergency and Stability of Explicit and Implicit Schemes in the Simulation of the Heat Equation

2021 ◽  
Vol 11 (10) ◽  
pp. 4468
Author(s):  
Franyelit Suárez-Carreño ◽  
Luis Rosales-Romero
Keyword(s):  
Heat Equation ◽  
Small Time ◽  
Simple Problem ◽  
Time Step ◽  
Temporal Dimension ◽  
Truncation Errors ◽  
Theoretical Predictions ◽  
Mesh Spacing ◽  
The One ◽  
Nicolson Scheme

Some strategies for solving differential equations based on the finite difference method are presented: forward time centered space (FTSC), backward time centered space (BTSC), and the Crank-Nicolson scheme (CN). These are developed and applied to a simple problem involving the one-dimensional (1D) (one spatial and one temporal dimension) heat equation in a thin bar. The numerical implementation in this work can be used as a preamble to introduce a method of solving the heat equation that can be implemented in problems in the area of finances. The results of implementing the software on very fine meshes (unidimensional), and with relatively small-time steps, are shown. Through mesh refinement, it was possible to obtain a better temperature distribution in the thin bar between a range of points. The heat equation was solved numerically by testing both implicit (CN) and explicit (FTSC and BTSC) methods. The examples show that the implemented schemes conform to theoretical predictions and that truncation errors depend on mesh, spacing, and time step.

scholarly journals Convergence Improved Lax-Friedrichs Scheme Based Numerical Schemes and Their Applications in Solving the One-Layer and Two-Layer Shallow-Water Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Xinhua Lu ◽  
Bingjiang Dong ◽  
Bing Mao ◽  
Xiaofeng Zhang
Keyword(s):  
High Resolution ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Small Time ◽  
Numerical Schemes ◽  
Time Step ◽  
Numerical Diffusion ◽  
First Order ◽  
Small Time Step ◽  
The One

The first-order Lax-Friedrichs (LF) scheme is commonly used in conjunction with other schemes to achieve monotone and stable properties with lower numerical diffusion. Nevertheless, the LF scheme and the schemes devised based on it, for example, the first-order centered (FORCE) and the slope-limited centered (SLIC) schemes, cannot achieve a time-step-independence solution due to the excessive numerical diffusion at a small time step. In this work, two time-step-convergence improved schemes, the C-FORCE and C-SLIC schemes, are proposed to resolve this problem. The performance of the proposed schemes is validated in solving the one-layer and two-layer shallow-water equations, verifying their capabilities in attaining time-step-independence solutions and showing robustness of them in resolving discontinuities with high-resolution.

scholarly journals Simultaneous Identification of Thermal Conductivity and Heat Source in the Heat Equation

2021 ◽  
pp. 1968-1978
Author(s):  
M. J. Huntul ◽  
M.S. Hussein
Keyword(s):  
Thermal Conductivity ◽  
Inverse Problem ◽  
Heat Source ◽  
Heat Equation ◽  
Mathematical Formulation ◽  
Approximate Solutions ◽  
Ill Posed ◽  
The One ◽  
Nicolson Scheme ◽  
Noisy Measurements

This paper presents a numerical solution to the inverse problem consisting of recovering time-dependent thermal conductivity and  heat source coefficients  in the one-dimensional  parabolic heat equation.   This  mathematical  formulation  ensures that the inverse problem  has a unique  solution.   However, the problem  is still  ill-posed since small errors  in the input data lead to a drastic  amount  of errors in the output coefficients.  The  finite  difference method  with  the Crank-Nicolson  scheme is adopted  as a direct  solver of the problem in a fixed domain.   The inverse problem is solved subjected to both exact and noisy measurements  by using the MATLAB  optimization  toolbox  routine  lsqnonlin , which is also applied to minimize the nonlinear  Tikhonov  regularization functional.  The thermal conductivity and heat source coefficients are reconstructed using heat flux measurements. The root mean squares error is used to assess the accuracy of the approximate solutions of the problem. A couple of  numerical  examples are presented to verify the accuracy and stability of the solutions.

The Dispersion in Finite Element Solutions to the One-Dimensional Heat Equation

1999 ◽  
Author(s):  
Ashley F. Emery ◽  
Walter Dauksher
Keyword(s):  
Finite Element ◽  
Heat Equation ◽  
Numerical Algorithm ◽  
Mesh Refinement ◽  
Matrix Formulation ◽  
Time Step ◽  
One Dimensional ◽  
The One ◽  
Capacitance Matrix

Abstract A method for evaluating the numerically introduced dispersion in finite element solutions to the one-dimensional heat equation is presented. The dispersion is quantified for linear and quadratic elements as a function of time step, mesh refinement and capacitance matrix formulation. It is demonstrated that an analysis of the dispersion is a useful tool in estimating the accuracy and in understanding the behavior of the numerical algorithm.

scholarly journals Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

2014 ◽  
Vol 16 (3) ◽  
pp. 817-840 ◽  
Author(s):  
E. Ferrer ◽  
D. Moxey ◽  
R. H. J. Willden ◽  
S. J. Sherwin
Keyword(s):  
Discontinuous Galerkin ◽  
Incompressible Flows ◽  
Projection Methods ◽  
Small Time ◽  
High Order ◽  
Time Step ◽  
Mesh Spacing ◽  
High Reynolds ◽  
Stability Limits ◽  
Lower Limits

AbstractThis paper presents limits for stability of projection type schemes when using high order pressure-velocity pairs of same degree. Two high orderh/pvariational methods encompassing continuous and discontinuous Galerkin formulations are used to explain previously observed lower limits on the time step for projection type schemes to be stable [18], when h- or p-refinement strategies are considered. In addition, the analysis included in this work shows that these stability limits do not depend only on the time step but on the product of the latter and the kinematic viscosity, which is of particular importance in the study of high Reynolds number flows. We show that high order methods prove advantageous in stabilising the simulations when small time steps and low kinematic viscosities are used.Drawing upon this analysis, we demonstrate how the effects of this instability can be reduced in the discontinuous scheme by introducing a stabilisation term into the global system. Finally, we show that these lower limits are compatible with Courant-Friedrichs-Lewy (CFL) type restrictions, given that a sufficiently high polynomial order or a mall enough mesh spacing is selected.

Time and space meshes adaptivity for the resolution of a semi-linear heat equation

2021 ◽  
pp. 1-10
Author(s):  
Nejmeddine Chorfi
Keyword(s):  
Parabolic Equation ◽  
Heat Equation ◽  
Order Of Convergence ◽  
Linear Parabolic Equation ◽  
Spectral Elements ◽  
Euler Scheme ◽  
Time Step ◽  
Implicit Euler Scheme ◽  
Linear Heat ◽  
Error Indicators

The aim of this work is to highlight that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal for a semi-linear parabolic equation discretized by an implicit Euler scheme in time and spectral elements method in space. The numerical results confirm the optimality of the order of convergence. The later is similar to the order of the error indicators.

scholarly journals Determination of the Height of Hard X-Ray Sources in the Solar Atmosphere by Measurement of Photospheric Albedo Photons

1975 ◽  
Vol 68 ◽  
pp. 239-241
Author(s):  
John C. Brown ◽  
H. F. Van Beek
Keyword(s):  
Solar Atmosphere ◽  
The Other ◽  
X Ray ◽  
Theoretical Predictions ◽  
Source Models ◽  
The One

SummaryThe importance and difficulties of determining the height of hard X-ray sources in the solar atmosphere, in order to distinguish source models, have been discussed by Brown and McClymont (1974) and also in this Symposium (Brown, 1975; Datlowe, 1975). Theoretical predictions of this height, h, range between and 105 km above the photosphere for different models (Brown and McClymont, 1974; McClymont and Brown, 1974). Equally diverse values have been inferred from observations of synchronous chromospheric EUV bursts (Kane and Donnelly, 1971) on the one hand and from apparently behind-the-limb events (e.g. Datlowe, 1975) on the other.

scholarly journals Farm Gate Prices for Non-Varietal Wine in Argentina: A Multilevel Comparison of the Prices Paid by Cooperatives and Investor-Oriented Firms

2017 ◽  
Vol 16 (1) ◽  
Author(s):  
Agustina Malvido Perez Carletti ◽  
Markus Hanisch ◽  
Jens Rommel ◽  
Murray Fulton
Keyword(s):  
Multilevel Regression ◽  
Data Set ◽  
Unique Data ◽  
Market Shares ◽  
Price Effects ◽  
Theoretical Predictions ◽  
The One ◽  
The Cost ◽  
Price Differences ◽  
Multilevel Regression Model

AbstractIn this paper, we use a unique data set of the prices paid to farmers in Argentina for grapes to examine the prices paid by non-varietal wine processing cooperatives and investor-oriented firms (IOFs). Motivated by contrasting theoretical predictions of cooperative price effects generated by the yardstick of competition and property rights theories, we apply a multilevel regression model to identify price differences at the transaction level and the departmental level. On average, farmers selling to cooperatives receive a 3.4 % lower price than farmers selling to IOFs. However, we find cooperatives pay approximately 2.4 % more in departments where cooperatives have larger market shares. We suggest that the inability of cooperatives to pay a price equal to or greater than the one paid by IOFs can be explained by the market structure for non-varietal wine in Argentina. Specifically, there is evidence that cooperative members differ from other farmers in terms of size, assets and the cost of accessing the market. We conclude that the analysis of cooperative pricing cannot solely focus on the price differential between cooperatives and IOFs, but instead must consider other factors that are important to the members.

scholarly journals Comparison of numerical schemes of river flood routing with an inertial approximation of the Saint Venant equations

RBRH ◽  
2018 ◽  
Vol 23 (0) ◽  
Author(s):  
Alice César Fassoni-Andrade ◽  
Fernando Mainardi Fan ◽  
Walter Collischonn ◽  
Artur César Fassoni ◽  
Rodrigo Cauduro Dias de Paiva
Keyword(s):  
Numerical Stability ◽  
Flood Routing ◽  
Computational Time ◽  
Numerical Schemes ◽  
Time Step ◽  
Simple Formulation ◽  
One Dimensional ◽  
Saint Venant Equations ◽  
Original Scheme ◽  
The One

ABSTRACT The one-dimensional flow routing inertial model, formulated as an explicit solution, has advantages over other explicit models used in hydrological models that simplify the Saint-Venant equations. The main advantage is a simple formulation with good results. However, the inertial model is restricted to a small time step to avoid numerical instability. This paper proposes six numerical schemes that modify the one-dimensional inertial model in order to increase the numerical stability of the solution. The proposed numerical schemes were compared to the original scheme in four situations of river’s slope (normal, low, high and very high) and in two situations where the river is subject to downstream effects (dam backwater and tides). The results are discussed in terms of stability, peak flow, processing time, volume conservation error and RMSE (Root Mean Square Error). In general, the schemes showed improvement relative to each type of application. In particular, the numerical scheme here called Prog Q(k+1)xQ(k+1) stood out presenting advantages with greater numerical stability in relation to the original scheme. However, this scheme was not successful in the tide simulation situation. In addition, it was observed that the inclusion of the hydraulic radius calculation without simplification in the numerical schemes improved the results without increasing the computational time.

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