Table 3-1Numerical solution of the Diffusion EquationTable 3-2Numerical solution of the Conduction EquationTable 3-3Numerical solution of the Conduction EquationTable 3-4Crank-Nicholson Solution of the Conduction EquationTable 3-5Fully-implicit Solution of the Conduction EquationTable 3-6Comparison of Results and Errors at time = tmax for Various MethodsUsing equation [3-36] for ρJacobi gives the following resultCollege of Engineering and Computer ScienceMechanical Engineering DepartmentEngineering Analysis NotesLarry Caretto March 24, 2009Numerical Analysis of Partial Differential EquationsSolution Properties for Finite-difference EquationsA numerical solution to an ordinary or a partial-differential equation should satisfy various properties. These are listed below.Consistency. A finite-difference expression is consistent with the differential equation if the truncation error approaches zero (ignoring roundoff) as the discrete steps (in space and time) approach zero. This is usually the case for finite-difference expressions. However, there are some algorithms, such as the DuFort-Frankel algorithm, in which the truncation error depends on the ratio (Δt/Δx)2, that are only conditionally consistent.Stability. A finite-difference equation should be stable. The errors in a stable finite-difference equation will not grow without bound as the solution progresses.Convergent. A convergent finite-difference expression tends to the exact solution of the differential equation as the grid size tends towards zero. According to the Lax Equivalence Theorem, a properly posed linear initial value problem and a finite difference approximation to it that satisfies the consistency condition will be convergent if it is stable.Physical reality. Solutions should produce physically realistic results. Densities should be positive. Temperature changes should not violate the second law of thermodynamics. This should be true for each node in the solution. This requirement applies not only to the numerical method, but to physical models for complex flow phenomena such as turbulence, combustion, gaseous radiation, and multiphase flow.Accuracy. There are many sources of error in numerical solutions. We have discussed truncation errors caused by the numerical approaches. Additional errors, known as iteration errors, are possible when approximate solutions to the finite-difference equations are obtained. Furthermore, inaccuracies can be introduced by poor physical models or assumptions. For example, the solution of a potential flow problem will have possible errors from truncation and iteration error. However, the assumption of potential flow can introduce errors as compared to theactual physical problem. However, these errors may be acceptable in some cases.Finite-difference methods and stability for the diffusion equationThe notes on the entitled “Introduction to Numerical Calculus”, referred to here as “introductory notes”, we applied finite-difference and finite-element methods to a simple ordinary differential equation. The extension of the finite-difference approach used there to partial differential equations is fairly straightforward. As an example of this, consider the following differential equation, known as the diffusion equation or the heat conduction equation.22xTtT[3-1]Jacaranda (Engineering) 3333 Mail Code Phone: 818.677.6448E-mail: [email protected] 8348 Fax: 818.677.7062The quantity  represents the thermal diffusivity in heat transfer and the diffusion coefficient in diffusion problems. We will call this term the diffusivity. The dependent variable T can be a general potential, although here we are using the usual symbol for temperature. Equation [3-1] has an open boundary in time; we do not have to specify the conditions at a future time. This equation is formally classified as a parabolic partial differential equation.We need an initial condition for the temperature at all points in the region. We can write a generalinitial condition as T(x,0) = T0(x). Similarly we can write arbitrary boundary conditions for the boundaries at xmin and xmax: T(xmin,t) = TL(t) and T(xmax,t) = TR(t). These conditions, as well as values for the geometry and the diffusivity must be specified before we can solve the differential equation.We can construct finite-difference grids in space and time. For simplicity, we will assume constant step sizes, Δt and Δx. We define our time and space coordinates on this grid by the integers n and i, so that we can writetixxandtnttin00[3-2]We can obtain a solution process, known as the explicit method, if we use a forward difference for the time derivative and a central difference for the space derivatives. This method is also called the forward-time-central-space (FTCS) method to emphasize the nature of the finite-difference approximations to the time and space derivatives in the diffusion equation. The forward-time and central-space approximations are given by the following equations.])[()(2)(2211221xOxTTTxTandtOtTTtTninininininini[3-3]These equations are modifications of equations [19] and [29] from the introductory notes on numerical calculus. In those notes we dealt with ordinary derivatives and needed only one subscript for the dependent grid variable. Here, in a multidimensional case, we have dependent variable, T, as a function of time and distance, T(x,t). Thus we define Tin = T(xi, tn). We use differences between variables like Tin, T ni+1, and Tni-1 to give differences in the x-direction that are used to compute spatial derivatives. The constant n superscript in the space derivative is an indication that we are holding time constant and varying distance. Similarly, the time derivatives are reflected in the finite-difference form by using variations in the superscript n while holding the subscript i constant.The forward difference for the time derivative is chosen over the more accurate central difference to provide a simple algorithm for solving this problem. There are other problems with the use of the central difference expression for the time derivative that we will discuss later.If we substitute the finite-difference expressions in [3-3] into equation [3-1], we get the following result.])(,[)(222111xtOxTTTtTTninininini[3-4]We have written the combined error term to