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methods for unsteady problem

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methods for unsteady problem
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  1  In computing unsteady  flows, we have a fourth  coordinate direction to consider: time    Just as with the space coordinates, time must be discretized  Essentially all unsteady solution methods advance in time in a step-by-step or “marching” manner  These methods are very similar to ones applied to initial value problems for ordinary differential equations (ODEs)  A brief review of such methods is given here   D R . AJMAL SHAH, PIEAS CH-06: METHODS FOR UNSTEADY PROBLEMS  2  Method involving the values of unknown at only two times  Explicit or forward Euler method  Implicit or backward Euler method  Midpoint rule  Trapezoid rule  Richardson extrapolation Two-Level Methods   Predictor-Corrector and Multipoint Methods    The predictor-corrector method  Runge-Kutta methods  Adams-Bashforth methods  Adams-Moulton methods  Leapfrog method  Crank-Nicolson method   D R . AJMAL SHAH, PIEAS METHODS FOR INITIAL VALUE PROBLEMS IN ODES  3 )}(,{ )( t t  f  dt t d       Consider the first order ordinary differential equation with an initial condition: t t t t t t t t t  nno   1121 .......  We have to find the solution ϕ  a short time Δ t  after the initial point at t o      11 )}(,{ 1 nnnn t t nnt t  dt t t  f  dt  dt d       oo t        )( n     ...... 21   D R . AJMAL SHAH, PIEAS TWO-   LEVEL METHODS  4 rule)Trapezoid(  Midpoint rule   provides the basis for Leapfrog   method  and  Trapezoid rule  provide the basis for Crank-Nicolson   method   D R . AJMAL SHAH, PIEAS TWO-   LEVEL METHODS   t t  f    nnnn   ),( 1      method)Euler forwardor Explicit ( t t  f    nnnn    ),(  111      method)Euler  backwardor Implicit ( t t  f   nnnn    ),(  21211      rule)Midpoint ( t t  f  t  f    nnnnnn    )],(),([ 21  111     

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Jul 23, 2017

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