College of Engineering and Computer Science Mechanical Engineering Department Engineering Analysis Notes Larry Caretto March 24 2009 Numerical Analysis of Partial Differential Equations Solution Properties for Finite difference Equations A 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 t x 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 the actual physical problem However these errors may be acceptable in some cases Finite difference methods and stability for the diffusion equation The 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 T 2T t x 2 Jacaranda Engineering 3333 E mail lcaretto csun edu Mail Code 8348 3 1 Phone 818 677 6448 Fax 818 677 7062 The 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 general initial 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 t and x We define our time and space coordinates on this grid by the integers n and i so that we can write tn t0 n t xi x0 i t and 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 finitedifference approximations to the time and space derivatives in the diffusion equation The forward time and central space approximations are given by the following equations T t n i T n 1 Ti n i O t t and 2T x 2 n i Ti n1 Ti n1 2Ti n O x 2 2 x 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 T in 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 Ti n 1 Ti n Ti n1 Ti n1 2Ti n O t x 2 2 t x 3 4 We have written the combined error term to show the order of the error for both x and t We have not written these as separate terms since O notation simply shows how the error depends on the power of the step size If we neglect the truncation error we can obtain a simple equation for the value of the temperature at the new time step Ti n 1 t Ti n1 Ti n 1 2Ti n Ti n x 2 3 5 We see that the grid spacings and the thermal diffusivity appear in a single term We can define this as a single term to save writing f t x 2 3 6 With this definition equation 3 5 may be written as follows Ti n 1 f Ti n1 Ti n1 1 2 f Ti n 3 7 With this finite difference equation each value of T at the new time step is related only to T values at the