Table 2-1Results using first-order forward differencesStep SizeErrorComparison of Errors in Finite-Difference (FDM) and Finite-Element (FEM) Solutions to Equation [2-33]College of Engineering and Computer ScienceMechanical Engineering DepartmentEngineering Analysis NotesLarry Caretto November 3, 2004Introduction to Numerical CalculusIntroductionBefore considering the numerical analysis of the differential equations for engineering analysis it is necessary to examine in a general way the methods used for the numerical analysis of differential equations. Any problem posed in terms of a differential equation must include a set of boundary conditions.The boundaries may be closed or open in a mathematical sense. If there is an open boundary, it is not necessary to specify boundary conditions on that open boundary. In physical problems an open boundary occurs most often when time is the variable and conditions need not be specified at a future time.Various kinds of boundary conditions can arise in engineering problems. If the dependent variable is specified at the boundary, the boundary condition is called a Dirichlet boundary condition or a boundary condition of the first kind. If the gradient of the dependent variable is specified, the boundary condition is called a Neumann boundary condition or a boundary condition of the second kind. When the boundary conditions are specified as a relationship between the dependent variable and its gradient, the boundary conditions are classified as being of the third kind or mixed.The three different types of boundary conditions can be illustrated in a heat transfer problem where we are solving a differential equation for temperature. A specified boundary temperature would be a Dirichlet boundary condition. A specified boundary heat flux, which is proportional to the temperature gradient by Fourier’s Law, would be a Neumann or second-kind boundary condition. Convection boundary conditions are mixed – or third-kind – boundary conditions. In such a boundary condition, the external convective heat flux is set equal to the internal conductive heat flux. This gives a mixed boundary condition such as hconv(T∞ - T) = -k∂T/∂x at x = 0.All numerical methods convert the differential equation and boundary conditions into a set of simultaneous linear algebraic equations. The differential equation provides an accurate description of the dependent variable at any points in the region where the equation applies. The numerical approach provides approximate numerical values of the dependent variable at a set of discrete points in the region. The values of the dependent variable at these points are found by solving the simultaneous algebraic equations.Two fundamentally different approaches are used to derive the algebraic equations from the differential equations: finite differences and finite elements. In the finite-difference approach, numerical approximations to the derivatives occurring in the differential equations are used to replace the derivatives at a set of grid nodes. In the finite-element approach, the region is dividedinto elements and an approximate behavior for the dependent variable over the small element is used. Both of these methods will be explored in these notes, following a discussion of some fundamental ideas.Engineering Building Room 1333 Mail Code Phone: 818.677.6448E-mail: [email protected] 8348 Fax: 818.677.7062Page 2 L. S. Caretto, November 3, 2004 Numerical MethodsAlthough much of the original work in numerical analysis used finite differences, many engineering codes currently used in practice use a finite element approach. The main reason for this is the ease with which the finite element method may be applied to irregular geometries. Some codes have used combination techniques that apply finite difference methods to grids generated for irregular geometries used by finite-element calculations.Finite-difference gridsIn a finite-difference grid, a region is subdivided into a set of discrete points. The spacing between the points may be uniform or non-uniform. For example, a grid in the x direction, xmin ≤ x≤ xmax may be written as follows. First we place a series of N+1 nodes numbered from zero to N in this region. The coordinate of the first node, x0 equals xmin. The final grid node, xN = xmax. The spacing between any two grid nodes, xi and xi-1, has the symbol Δxi. These relations are summarized as equation [2-1].x0 = xmin xN = xmax xi – xi-1 = Δxi[2-1]A non-uniform grid, with different spacing between different nodes, is illustrated below.●---------●------------●------------------●---~ ~-------●----------●------● x0 x1 x2 x3 xN-2 xN-1 xNFor a uniform grid, all values of Δxi are the same. In this case, the uniform grid spacing, in a one-dimensional problem is usually given the symbol h. I.e., h = xi – xi-1 for all values of i.In two space dimensions a grid is required for both the x and y, directions, which results in the following grid and geometry definitions, assuming that there are M+1 grid nodes in the y direction.x0 = xmin xN = xmax xi – xi-1 = Δxiy0 = ymjn yM = ymay yj – yj-1 = Δyj[2-2]For a three-dimensional transient problem there would be four independent variables: the three space dimensions, x, y and z, and time. Each of these variables would be defined at discrete points, i.e.x0 = xmin xN = xmax xi – xi-1 = Δxiy0 = ymjn yM = ymax yj – yj-1 = Δyj[2-3]z0 = zmin zK = zmax zk – zk-1 = Δzkt0 = tmin tL = tmax tn – tn-1 = ΔtnAny dependent variable such as u(x,y,z,t) in a continuous representation would be defined only atdiscrete grid points in a finite-difference representation. The following notation is used for the set of discrete values of dependent variables.),,,(nkjinijktzyxuu [2-4]For steady-state problems, the n superscript is omitted. For problems with only one or two space dimensions two or one of the directional subscripts may be omitted. The general use of the notation remains. The subscripts (and superscript) on the dependent variable represent a particular point in the region, (xi, yj, zk, tn) where the variable is