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Unformatted text preview: in order to show the convergence
as both explicit and implicit schemes are linear. We will discuss the stability of explicit
scheme using Neumann approach. On the same lines, the stability of implicit scheme
can also be proved. However, the solution by FD, when there is discontinuity in the initial conditions, is not
accurate as the propagation of discontinuity problem in the stability. However, the
problem with no discontinuities can be solved satisfactorily & efficiently by convergent
and stable FD methods with rectangular grids. 2u 2u ,
t 2
x 2 Example: Solve the wave equation 0 x 1, t0 using Implicit Method subject to the conditions:
u (0, t ) 0 u (1, t ) u
( x, 0) 0
t u ( x, 0) Cos x, Solution: By implicit method, equation is given by
+(1+ , ) , = , + +( 1 , ) + (2 , ) , + , + , , Putting r = 1/2
⇒ + , – , , = , + , ,  + , + + , (1) , B.C. u(0, t) = 0 = u(1, t)
⇒ u(0, 1) = u(0, 2) = u(0, 3) = …. = 0 = u(0, 0) & u(5, 0) = u(5, 1) = u(5, 2...
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This note was uploaded on 04/07/2014 for the course MATH 545 taught by Professor Prof.ramabhargava during the Spring '14 term at Indian Institute of Technology, Roorkee.
 Spring '14
 Prof.RamaBhargava
 Differential Equations, Equations, Derivative, Partial Differential Equations

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