Higher Order Finite Difference Schemes for the Heat Equation

Higher Order Finite Difference Schemes for the Heat Eq (Part 1)

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Higher Order Finite Difference Schemes for the Heat Equation: A comparative Analysis

Abstract

In this post, we studied one and two-dimensional parabolic PDEs, such as heat equations. There are numerous applications of such differential equations in physical models and engineering models. Heat equation governs a variety of physical phenomena, for instance; heat transfer in solids. The discretization of the heat equation is using two points and three points finite difference approximation.

Higher Order Finite Difference Schemes for the Heat Equation

We have solved the heat equation by using three points forward in time and backward in time, central in space and crank Nicolson; and the Adams-Bashforth method. The heat conduction in a slab of material with a semi-infinite strip of length L with Dirichlet type BCs considered in this post. We also discuss its stability.

Introduction

In the field of fluid dynamics as well as in solid mechanics almost all problems come out as systems of PDE’s. PDEs have many applications in physics, for example; the light and sound waves; heat propagation; elasticity; fluid flow; electrodynamics; electrostatics; etc.

Higher Order Finite Difference Schemes for the Heat Equation

Many important and physical problems are intensively modeled by linear and non linear PDEs. For example, modeling of flow of traffic on a busy road, flow of blood through flexible wall tube and particular cases of general theories of hydraulics and dynamics of gaseous materials. The momentum, continuity equation and energy equation these are also modeled as PDEs. Whole fluid mechanics based on these three fundamental laws.

Higher Order Finite Difference Schemes for the Heat Equation

Mathematicians have been quite interested in dealing heat equation from the 19th century. Heat equation has been derived using Fourier’s laws and phenomenon of energy conservation. P. Nicolson and J. Crank in 1947 observed a new dimension for the numerical solution by finite difference method that was unconditionally unstable. Many researchers have discussed the numerical solution of heat equation. Ames gave a better mathematical progress of the finite difference technique. Hoffman studied parabolic, hyperbolic and elliptic PDE’s. Mayers and Morton discussed briefly the upwind scheme. Ortega has adopted an applied approach which incorporates implementation issues.

Higher Order Finite Difference Schemes for the Heat Equation

Heat equation or generally the energy equation arises in numerous Mechanical fluid problems. The numerical solution of the heat equation has been extensively studied in various cases, as demonstrated by Recktenwald. Among numerical techniques, the finite difference method is one of the simplest and most commonly used for solving the heat equation. This numerical scheme is directly influence the order of convergence by using the different number of points.

PDEs

A PDE is an equation that involves two or more independent variables. This relationship of unknown function w(t,x) w.r.t time t and space x is denoted by

\dfrac{\partial w}{\partial t} and \dfrac{\partial w}{\partial x} respectively. For instance

(1)   \begin{equation*} \dfrac{\partial w}{\partial t}= \alpha \dfrac{\partial^2 w}{\partial x^2} \end{equation*}


where \alpha ={\frac {k}{c_{p}\rho }}.

Higher Order Finite Difference Schemes for the Heat Equation

A PDE with power of unknown function and all partial derivatives involved is one, are called linear PDE. The PDE xw_{xx}+yw_{yy}=0 is linear. Coefficients of dependent function and coefficients of all the partial derivative generally are constants terms or independent functions. For example, the differential equation w_x+w^2w_{xx}=x^2 is nonlinear.

Second order linear PDE is given by

(2)   \begin{equation*} Aw_{xx} + Bw_{xy} + Cw_{yy} + Dw_x + Ew_y + Gw = f(x,y),\end{equation*}

where A to G are constant or functions of x and y. The appropriate form known as
Euller equation is

(3)   \begin{equation*} Aw_{xx} + Bw_{xy} + Cw_{yy}=0\end{equation*}

The equation (2) can be classified in the following three categories;

Parabolic Equation

This equation will called parabolic if B^2 - 4AC= 0 for all (x,y). For example, heat equation w_{t} = \alpha w_{xx} , is a parabolic equation which explains the heat distribution for a given domain over time.

Elliptic Equation

This equation will called Elliptic if B^2 - 4AC < 0 for all (x,y). For example, Laplace equation w_{xx}+w_{yy} =0

Hyperbolic Equation

This equation will be called hyperbolic if B^2 - 4AC > 0 for all (x,y). For example, wave equation w_{tt} = c^2 w_{xx} where c is the speed of the wave. Wave equation is a 2nd order hyperbolic PDE and it used to describe the waves, as occur in physics such as light waves, sound waves.

Parabolic PDE’s and BCs

Compared to the general solution, the particular solution of a PDE is more valuable because it directly addresses the specific conditions of a given problem. For a given PDE, which controls the physical phenomenon along with mathematical behavior for bounded domain, the dependent function w is generally defined at boundary of domain. There are many BCs that can be applied on PDE’s; such as Dirichlet, Robin and Neumann BCs.

Dirichlet BCs

The BCs that defines the value of function itself is Dirichlet BCs or First type of BCs. Here the function is generally defined in given domain at the boundary. Suppose a strip having length L, then w(0)=w_1 and w(L)= w_2 may regarded as BCs, where w_1 and w_2 are constants.

For the heat equation we can write

(4)   \begin{equation*} \frac{ \partial w}{\partial t}=\alpha \frac{\partial^2 w}{\partial x^2},\end{equation*}

The Dirichlet BCs for equation (4) can be written as

\begin{center}
w(0,t)=f_1 and w(L,t)=f_2
\end{center}

where f_1 and f_2 are constants.

Neumann BCs

The condition of boundary which specifies the values in which the derivative of a solution is applied within the boundary of the domain is called Neumann BCs or second type BCs. The Neumann BCs for equation (4) can be written as

\begin{center}
w_x(0,t)=g_1 and w_x(L,t)=g_2
\end{center}

where g_1 and g_2 are constants.

Robin BCs

Robin BCs are those conditions which are weighted combination of the Dirichlet BCs and Neumann BCs. Here the dependent variable w and the normal form \frac{\partial w}{\partial x} in a linear combination is proposed on boundary. The Robin BCs for equation (4) can be written as

    \[aw(0,t)-bw_x(0,t)=h_1 \quad \text{and} \quad aw(L,t)+bw_x(L,t)=h_2\]

where a,b,h_1 and h_2 are constants.

Taylor Series

A series of a function which is infinite, in terms of the derivatives of function evaluated at a in the neighborhood of a point a is called Taylor Series.

Mathematically,

    \[w_{i+1}=w_i+\Delta x \dfrac{\partial w}{\partial x}\big|{x_i}+\dfrac{\Delta x^2}{2} \dfrac{\partial^2 w}{\partial x^2}\big|{x_i} +\dfrac{\Delta x^3}{3!} \dfrac{\partial^3 w}{\partial x^3}\big|_{x_i}+\cdots\]

Analytical Solution

Analytical solution is the exact solution of a problem. Sometimes it’s very hard to find analytical solution, then we use numerical solution.

Numerical Solution

The solution of a problem that calculated by taking approximations on that problem. For the numerical solution, we usually use computer programming.

Error

Mathematical information about the accuracy of calculated numerical solution.

    \[error= \parallel w_e-w_n\parallel\]


w_n is the numerical solution of a given problem.

Truncation Error

A small residuum is obtained when we substitute the exact solution w of the differential equation in the discretization formulation. This residuum is called truncation error.

Order of a Scheme

Order of a scheme tells us about the accuracy of the scheme, higher order scheme gives more accurate results.

Solution Methodology

We calculated the solution of the heat equation both analytically and numerically. We used the method of separation of variables to obtain the analytical solution. The solution is estimate by converting this parabolic PDE into a set of ODE’s. For the discretization of heat equation we use the finite difference approximations. The schemes used are FTCS (or explicit Euler scheme), BTCS (or implicit Euler scheme), Crank Nicolson and Adams Bashforth schemes. In next post the stability of the schemes also discussed.

Higher Order Finite Difference Schemes for the Heat Equation

In this post the explicit and implicit schemes. Explicit schemes are so simple to derive and implement. Implicit schemes are difficult for implementation to any problem as compared to explicit schemes. Hardest part for implicit schemes is to find the solution of resulting linear system of equations. On the other hand, there is no need to solve the linear system of equations in explicit schemes. Explicit schemes typically have stability restrictions or more probably are unstable. Implicit scheme are always unconditionally stable. In this thesis, forward in time centered in space is explicit scheme but backward in time centered in space and Crank-Nicolson schemes are implicit schemes. Finite difference method is easy as compared to other methods i.e. Finite element method or finite volume method.

Higher Order Finite Difference Schemes for the Heat Equation

Current chapter involves few primary definitions and summarize some prerequisites regarding finite difference. It provides some basic information about the under consideration of solution of heat equation and also throws light on the FTCS, BTCS and Crank-Nicolson schemes. Analytical and numerical results are verified. The last chapter concludes the current work and provides possible directions for future research work.

Higher Order Finite Difference Schemes for the Heat Equation

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Higher Order Finite Difference Schemes for the Heat Equation

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