# Implicit finite difference method heat transfer

In numerical analysisfinite-difference methods FDM are discretizations used for solving differential equations by approximating them with difference equations that finite differences approximate the derivatives. FDMs convert a linear ordinary differential equations ODE or non-linear partial differential equations PDE into a system of equations that can be solved by matrix algebra techniques. First, assuming the function whose derivatives are to be approximated is properly-behaved, by Taylor's theoremwe can create a Taylor series expansion.

We will derive an approximation for the first derivative of the function "f" by first truncating the Taylor polynomial:. The error in a method's solution is defined as the difference between the approximation and the exact analytical solution.

The two sources of error in finite difference methods are round-off errorthe loss of precision due to computer rounding of decimal quantities, and truncation error or discretization errorthe difference between the exact solution of the original differential equation and the exact quantity assuming perfect arithmetic that is, assuming no round-off.

To use a finite difference method to approximate the solution to a problem, one must first discretize the problem's domain. This is usually done by dividing the domain into a uniform grid see image to the right. This means that finite-difference methods produce sets of discrete numerical approximations to the derivative, often in a "time-stepping" manner.

An expression of general interest is the local truncation error of a method. Typically expressed using Big-O notationlocal truncation error refers to the error from a single application of a method. The remainder term of a Taylor polynomial is convenient for analyzing the local truncation error. A final expression of this example and its order is:. This means that, in this case, the local truncation error is proportional to the step sizes.

The quality and duration of simulated FDM solution depends on the discretization equation selection and the step sizes time and space steps. The data quality and simulation duration increase significantly with smaller step size. Large time steps are useful for increasing simulation speed in practice. However, time steps which are too large may create instabilities and affect the data quality.

The von Neumann and Courant-Friedrichs-Lewy criteria are often evaluated to determine the numerical model stability. The Euler method for solving this equation uses the finite difference quotient. The last equation is a finite-difference equation, and solving this equation gives an approximate solution to the differential equation. Consider the normalized heat equation in one dimension, with homogeneous Dirichlet boundary conditions.

One way to numerically solve this equation is to approximate all the derivatives by finite differences.Search for more papers by this author. Skip to main content. Volume 18, Issue 2. No Access. Thomas H.

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Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific problems. It only takes a minute to sign up. I am trying to model heat conduction within a wood cylinder using implicit finite difference methods. The general heat equation that I'm using for cylindrical and spherical shapes is:. Boundary conditions include convection at the surface. For more details about the model, please see the comments in the Matlab code below.

Running the above code gives a temperature profile at the center and surface of the wood cylinder:. As you can see from this plot, for some reason the center and surface temperatures rapidly converge at the 2 min mark which isn't correct. You only have boundary conditions for temperature which are the only boundary conditions you need.

It seems like you're using some sort of two-stage implicit predictor-corrector-like scheme to integrate the equations for temperature, where you do the following:. Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. Matlab solution for implicit finite difference heat equation with kinetic reactions Ask Question. Asked 6 years, 7 months ago. Active 3 years, 10 months ago. Viewed 8k times. Any suggestions on how to fix this or create a more efficient way to solve the problem? A suggestion for asking questions: please try to pick out short descriptions of the numerical methods you use and put them up front.

As you can see in my answer, I had to do a lot of digging through your code to figure out what you did. Things like "backward Euler" and "I have some other equations in my model, too" are helpful to know. Within each stage, you advance the species densities using explicit Euler. I see a couple potential problems with this scheme: Since temperature is really coupled to the wood and char densities, you're introducing what is called a physics splitting error that could be causing the numerical artifacts you've mentioned. Shrinking your time step will reduce this error.

Chemical source terms are sometimes stiff. You're integrating them with explicit Euler, which I wouldn't think to do intuitively based on the problems I studydue to stability issues.

8.2.5-PDEs: Implicit Finite Divided Difference for Parabolic PDEs

However, for most of your problem, there doesn't seem to be any great instability, and any instabilities you have are damped out, so maybe that's not an issue. Combining explicit and implicit methods in this way usually limits accuracy to first or second-order depending on splittingunless you use IMEX implicit-explicit methods. Your biggest problem is rolling your own time integrator, so you have no accuracy control. Or rather, your only way to control accuracy is to shrink the time step, look at your solution, and see if the new solution is more accurate.

Here's what I would do: Discretize your equations in space, and solve all of them simultaneously for now. If you have twenty points in the radial direction, and five state variables in the continuous formulation of your PDE, you'll only have state variables in your right-hand side.YASK--Yet Another Stencil Kit: a domain-specific language and framework to create high-performance stencil code for implementing finite-difference methods and similar applications. Pure Julia implementation of the finite difference frequency domain FDFD method for electromagnetics.

Solve the 1D forced Burgers equation with high order finite elements and finite difference schemes. Python package for the analysis and visualisation of finite difference fields. Finite-Difference Approximations to the Heat Equation. Solving partial differential equations using finite difference methods on Julia. A Python library for the development and analysis of finite-difference models and discrete dynamical systems.

Solving the Porous medium equation with a range of numerical finite difference methods. A web app solving Poisson's equation in electrostatics using finite difference methods for discretization, followed by gauss-seidel methods for solving the equations. Dirichlet conditions and charge density can be set. Fortran code to solve a two-dimensional unsteady heat conduction problem. Finite difference method in 2D; lecture note and code extracts from a computational course I taught. Source code for the course IN Numerical methods for partial differential equations at University of Oslo.

## 1D Heat Conduction using explicit Finite Difference Method

Solving the advection differnetial equation using the finite diference method. Finite difference method in 1D; lecture note and code extracts from a computational course I taught. Option pricing using the Binomial-tree, Monte Carlo method and Partial differential equation. Add a description, image, and links to the finite-difference-method topic page so that developers can more easily learn about it.

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The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I am trying to model heat conduction within a wood cylinder using implicit finite difference methods. The general heat equation that I'm using for cylindrical and spherical shapes is:. Boundary conditions include convection at the surface. For more details about the model, please see the comments in the Matlab code below.

Running the above code gives a temperature profile at the center and surface of the wood cylinder:. As you can see from this plot, for some reason the center and surface temperatures rapidly converge at the 2 min mark which isn't correct. It looks like you are using a backward Euler implicit method of discretization of a diffusion PDE. A more accurate approach is the Crank-Nicolson method. Both methods are unconditionally stable. The introduction of a T-dependent diffusion coefficient requires special treatment, best probably in the form of linearization, as explained briefly here.

It would be useful to identify stability criteria to ensure that the time and distance step lengths are appropriate following introduction of T-dependent coefficients. Note that matlab offers a PDE toolbox which might be useful to you, although I have not checked how you might use it in detail. Learn more. Matlab solution for implicit finite difference heat equation with kinetic reactions Ask Question.

Asked 6 years, 7 months ago. Active 6 years, 6 months ago. Viewed 5k times. Any suggestions on how to fix this or create a more efficient way to solve the problem?

Active Oldest Votes. Buck Thorn Buck Thorn 4, 2 2 gold badges 13 13 silver badges 25 25 bronze badges. I'll look into implementing your suggestions and see if that helps. To be continued I believe you are right in your suggestions. The fact that I'm varying the thermal properties requires me to use the form of the heat equation which incorporates variable thermal conductivity.

Authors: Norhashidah Hj. Mohd AliTeng Wai Ping. Keywords: Finite Differenceexplicit group methodHelmholtz equationrotated gridstandard grid. Authors: Yasser ElhenawyM. Abd ElkaderGamal H. A theoretical study of a humidification dehumidification solar desalination unit has been carried out to increase understanding the effect of weather conditions on the unit productivity.

A humidification-dehumidification HD solar desalination unit has been designed to provide fresh water for population in remote arid areas. It consists of solar water collector and air collector; to provide the hot water and air to the desalination chamber. The desalination chamber is divided into humidification and dehumidification towers.

The circulation of air between the two towers is maintained by the forced convection. A mathematical model has been formulated, in which the thermodynamic relations were used to study the flow, heat and mass transfer inside the humidifier and dehumidifier. The present technique is performed in order to increase the unit performance. Heat and mass balance has been done and a set of governing equations has been solved using the finite difference technique. The unit productivity has been calculated along the working day during the summer and winter sessions and has compared with the available experimental results.

The average accumulative productivity of the system in winter has been ranged between 2. In this paper, the formulation of a new group explicit method with a fourth order accuracy is described in solving the two dimensional Helmholtz equation.

The formulation is based on the nine-point fourth order compact finite difference approximation formula. The complexity analysis of the developed scheme is also presented. Several numerical experiments were conducted to test the feasibility of the developed scheme.

Comparisons with other existing schemes will be reported and discussed. Preliminary results indicate that this method is a viable alternative high accuracy solver to the Helmholtz equation. Keywords: Finite Differenceexplicit group methodfive-point formulanine-point formulaHelmholtz equation. Authors: Ahmad Izani Md. The generalized wave equation models various problems in sciences and engineering. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline for the approximate solution of wave equation is developed.

The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Von Neumann stability analysis is used to analyze the proposed method.

Two problems are discussed to exhibit the feasibility and capability of the method. The absolute errors and maximum error are computed to assess the performance of the proposed method. The results were found to be in good agreement with known solutions and with existing schemes in literature.

AliSam Teek Ling. We investigate the formulation and implementation of new explicit group iterative methods in solving the two-dimensional Poisson equation with Dirichlet boundary conditions. The methods are derived from a fourth order compact nine point finite difference discretization.

The methods are compared with the existing second order standard five point formula to show the dramatic improvement in computed accuracy.

Numerical experiments are presented to illustrate the effectiveness of the proposed methods.

### Finite Difference Related Publications

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