Solution of Differential Equation by Finite Difference Method

Transcription

1 NUMERICAL ANALYSIS University of Babylon/ College of Engineering/ Mechanical Engineering Dep. Lecturer : Dr. Rafel Hekmat Class : 3 rd B.Sc Solution of Differential Equation by Finite Difference Method What is a Differential Equation? An equation that consists of derivatives is called a differential equation. Differential equations have applications in all areas of science and engineering. Mathematical formulation of most of the physical and engineering problems leads to differential equations. Types of Differential Equations a- Ordinary Differential Equations (ODE) b- Partial Differential Equations (PDE) An ordinary differential equation is that in which all the derivatives are with respect to a single independent variable. Examples of ordinary differential equations include Ordinary differential equations are classified in terms of order and degree. Order of an ordinary differential equation is the same as the highest derivative and the degree of an ordinary differential equation is the power of highest derivative. Thus the differential equation, is of order 3 and degree 1, whereas the differential equation is of order 1 and degree. 1

2 PDEs are differential equations containing two or more independent variables, so the derivatives are partial derivatives. For example,given a function of two variables (x, y) f(x, y), the partial derivatives with respect to x and y are f x and f y PDE example I: Laplace equation The Laplace equation is a second ordered PDE appearing for example in Fluid Mechanics (potential flows) and Electromagnetics (electromagnetic field in a charge free region): where represents, for example, the velocity potential or the temperature. PDE example II: Heat equation The heat equation is a second order PDE describing how temperature T diffuses through a medium of thermal diffusivity α: PDE example III: Wave equation The wave equation describes how the sound pressure p propagates at speed c through a medium (at rest), and plays a role in acoustics, fluid mechanics, and quantum mechanics: Classification of Partial Differential Equations For analyzing the equations for fluid flow problems, it is convenient to consider the case of a second-order differential equation given in the general form as: In the coefficients A, B, C, D, E and F are either constants or functions of only (x, y) (do not contain φ or its derivatives), it is said to be a linear equation, otherwise it is a non-linear equation. If G =, the aforesaid equation is homogeneous, otherwise it is non-homogeneous.

3 Again for the above mentioned equation if B 4AC =, the equation is parabolic if B 4AC <, the equation is elliptic if B 4AC >, the equation is hyperbolic Example: Classify the following linear second order partial differential equations (PDEs) with solution u(x,y)in the xy-plane. 3

4 Example: Consider the one-dimensional damped wave equation 9uxx= utt+ 6ut Initial and Boundary Conditions Most PDEs have an infinite number of admissible solutions. Thus the PDE alone is not sufficient to get a unique solution. Usually some boundary conditions and initial conditions are required. For the heat equation the simplest boundary conditions are fixed temperatures at both ends: Where l is the length of the rod and h1(t) the temperature at the first end and h(t) the temperature at the second end. The initial conditions specify an arbitrary initial temperature distribution inside the rod: What is the finite difference method? The finite difference method is used to solve ordinary differential equations that have conditions imposed on the boundary rather than at the initial point. These problems are called boundary-value problems. Steps of finite difference solution: Divide the solution region into a grid of nodes, Approximate the given differential equation by finite difference equivalent, 4

5 Solve the differential equations subject to the boundary conditions and/or initial conditions. First Derivative Approximations The derivative of a given function f(x) can be approximated as: df ( x ) f ( x x) f ( x ) dx x df ( x ) f ( x ) f ( x x) dx x (1) () df x f x x f x x ( ) ( ) ( ) (3) dx x Forward Difference Formula Backword Difference Formula Central Difference Formula Second Derivative Approximation f f ( x x) f ( x ) f ( x x) x ( x) Example: Solve f 4 f, x 1. x Subject to the boundary conditions: Assume x=.5 f () f (1) 1 Solution: Discretization of the equation: f ( i 1) f ( i) f ( i 1) 4 ( ) f i ( x) f () f (1) f () 4 f (1) (.5) Since there is one unknown, one equation is enough to find this unknown. 5

6 If x=.5 f (1) f () f () f (1) 1 f () f (1) f () 4 f (1) (.5) f (3) f () f (1) 4 f () (.5) f (4) f (3) f () 4 f (3) (.5) f(1), f() and f(3) are unknowns. Therefore, three equations are written to find three unknowns. Example: Consider the diffusion equation: k t x Where k is a constant. Discretized equation: Where ( i, j 1) ( i, j) ( i 1, j) ( i, j) ( i 1, j) k t ( x) x i x, i,1,,..., n t j t, j,1,,..., n Forward difference for t and central difference for x is used. t r k( x) We can write: ( i, j 1) r ( i 1, j) (1 r) ( i, j) r ( i 1, j) This equation can be calculated in terms of boundary and initial conditions. This explicit formula can be used to find (x,t+ )explicitly in terms of (x,t) Example: The deflection in a simply supported beam with a uniform load q and a tensile axial load T is given by ` where x:location along the beam (in) T:Tension applied (lbs) E:Young s modulus of elasticity of the beam (psi) 6

7 I:Second moment of area (in4) q:uniform loading intensity (lb/in) L:Length of beam (in) Given, Find the deflection of the beam at x=5". Use a step size of Δx=5 and approximate the derivatives by central divided difference approximation. Solution: Substituting the given values, Approximating the derivative y at node i by the central divided difference approximation x We can rewrite the equation as: Since Δx=5, we have 4 nodes as given in Figure below Writing the equation at each node, we get 7

8 Equations above are 4 simultaneous equations with 4 unknowns and can be written in matrix form as Solving the equations we get, Example Take the case of a pressure vessel that is being tested in the laboratory to check its ability to withstand pressure. For a thick pressure vessel of inner radius a and outer radius b, the differential equation for the radial displacement u of a point along the thickness is given by The inner radius a= 5, and the outer radius b= 8, and the material of the pressure vessel is ASTM A36 steel Divide the radial thickness of the pressure vessel into 6 equidistant nodes, and find the radial displacement profile At node i along the radial thickness of the pressure vessel 8

9 Let us break the thickness, b-a, of the pressure vessel into n+1 nodes, that is is node and is node. That r=a is nodes i= and r=b is node i=n. That means we have n+1 unknowns. We can write the above equation for nodes 1,., n-1. This will give us n-1 equations. At the edge nodes, i= and i=n, we use the boundary conditions of We have been asked to do the calculations for n=5, that is a total of 6 nodes. This gives 9

10 Writing the above Equations in matrix form gives After solving Explicit versus implicit Finite Difference Schemes During the lecture we solved the transient (time-dependent) heat equation in 1D In explicit finite difference schemes, the temperature at time n+1 depends explicitly on the temperature at time n. The explicit finite difference discretization of above equation is This can rearranged in the following manner (with all quantities at time n+1 on the left-handside and quantities at time n on the right-hand-side) Since we knowt n n i+1, T i and T n i 1, we can compute T n i. The major advantage of explicit finite difference methods is that they are relatively simple and computationally fast. However, the main drawback is that stable solutions are obtained only when In implicit finite difference schemes, the spatial derivatives T are evaluated (at least partially) x at the new time step. The simplest implicit discretization of heat equation in 1D is This can be rearranged so that unknown terms are on the left and known terms are on the right Where s = K t. The main advantage of implicit finite difference methods is that there are x 1

11 no restrictions on the time step, which is good news if we want to simulate geological processes at high spatial resolution. Taking large time steps, however, may result in an inaccurate solution. Solving an implicit finite difference scheme We solve the transient heat equation on the domain l/ x l/ with the following boundary conditions with the initial condition As usual, the first step is to discretize the spatial domain with nx finite difference points. The implicit finite difference discretization of the temperature equation is and the known right-hand-side vector rhs is 11