Numerical Solutions to Ordinary Differential Equations

In this subsection, some commonly used numerical schemes that involve difference equations to solve ordinary differential equations are presented along with their stability characteristics. Simple examples to illustrate the effects of step size on the convergence of numerical methods are shown. A simple discretization of the first-order linear differential equation 

The solution of the differential equation approaches zero as x approaches infinity. The solution of the difference equation clearly depends on h, and it approaches zero only if hX l. For hX 1, the solution of the difference equation oscillates between very large positive and negative values for large n. Thus, the solution of the difference equation converges only for values h that satisfy hX l. For a nonlinear equation, 

This example illustrates that the step size, h, plays a crucial role in the convergence of the numerical solution of a differential equation. Depending on the equation to be solved and the method of discretization, h can be adjusted to obtain convergence of a numerical solution. 

A general method of solving a differential equation is to divide the interval of integration into n subintervals and then approximate the derivative by a one-step difference quotient, as in the Euler method or A -step difference quotient  

The order of a method is the integer p such that the difference between the term on the right-hand side of Equation (2.104) and y is proportional to The solution of the difference equation converges to that of the differential equation if the difference between the two solutions vanishes as the step size, h, approaches zero. A difference scheme is said to be zero-stable if the numerical solution of the equation ior fix,y) = 0 is zero. 

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