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Chapter 8 Elliptic Equation

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8-1 General Remarks The governing equations in fluid mechanics and heat transfer can be reduced to elliptic form for particular applications. Such examples are the steady-state heat conduction equation, velocity potential equation for incompressible, inviscid flow, and the stream function equation. Typical elliptic equations in a two-dimensional Cartesian system are Laplace’s equations, and Poisson’s equation

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8-2 Finite Difference Formulations (1) Five-point formula---central difference Laplace’s equations: F.D. which is accurate to

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8-2 Finite Difference Formulations (2) General form for five-point formula

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8-2 Finite Difference Formulations (3) From eqs. (1) and (2), we can get

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8-2 Finite Difference Formulations (4) In order to explore various solution procedures, first consider a square domain with Dirichlet B.C.s. For instance, a simple 6x6 grid system subject to the following B.C.s. is considered: x=0u=u 2, y=0 u=u 1 x=Lu=u 4, y=H u=u 3

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8-2 Finite Difference Formulations (5) The interior grid points produces sixteen equations with sixteen unknowns. The equations are:

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8-2 Finite Difference Formulations (6) Solution algorithms: (1) The Gauss-Seidel Iteration Method (point-by-point iteration method): (a) The finite difference equation is given by (b) For the computation of the first point, say (2,2), it follows that

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8-2 Finite Difference Formulations (7) (c) For point (3,2), one has (d) The general formulation is

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8-2 Finite Difference Formulations (8) (2) SOR Method: or What is the optimum value of ω? One such relation, for the solution of elliptic equations in a rectangular domain subject to Dirichlet B.C.s with constant step size, is

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8-2 Finite Difference Formulations (9) (3) Line-by-line iteration method: (a) In this formulation, it results in three unknowns at points (i-1,j), (i,j), and (i+1,j). It becomes (b) This equation, applied to all i at constant j, results in a system of linear equations which, in compact form, has a tridiagonal matrix coefficient. The solution of each row at constant j can be solved by TDMA method.

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8-2 Finite Difference Formulations (10) (c) Grid points employed in the line-by line iteration method.

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8-2 Finite Difference Formulations (11) (4) Line-by line SOR method: There is no simple way to determine the value of optimum ω. In practice, trial and error is used to compute ω opt for a particular problem.

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8-2 Finite Difference Formulations (12) (5) The Alternating Direction Implicit (ADI) Method: (a) An iteration cycle is considered complete once the resulting tridiagonal system is solved for all rows and then followed by columns, or vice versa. It follows

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8-2 Finite Difference Formulations (13) (b) Grid points used in ADI method:

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8-2 Finite Difference Formulations (14) (c) ADI with SOR Method:

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Chapter 9 Parabolic Partial Differential Equation

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9-1 General Remarks Equations of motion in fluid mechanics are frequently reduced to parabolic formulations. Boundary layer equations are examples of such formulations. In addition, the unsteady heat conduction equation is also parabolic.

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9-2 Finite Difference Formulations (1) A typical parabolic second-order PDE is the unsteady heat conduction equation, which is considered first in one-space dimension. It has the following form (1) FTCS (forward time/central space) method: (i) is expressed by a forward difference approximation which is of order :

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9-2 Finite Difference Formulations (2) (ii) Using the second-order central differencing of order for the diffusion term, eq. (8-1) can be approximated by (iii) Eq. (8-2) is also called explicit formulation, which is of order. It will be shown that the solution is stable for

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9-2 Finite Difference Formulations (3) (2) BTCS (backward/central space) method: (i) The above equation can be solved by TDMA.

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9-2 Finite Difference Formulations (4) (ii) Eq. (8-3) is defined as being implicit, since more than one unknown appears in the finite difference equation. As a result, a set of simultaneous equations needs to be solved, which require more computation time per time step. Implicit methods greater advantage on the stability of the finite difference equations, since most are unditionally stable. Therefore, a larger step size in time is permitted.

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9-2 Finite Difference Formulations (5) (3) CTCS (central time/central space) method: (The Crank- Nicolson method) (i) If the diffusion term of eq. (8-1) is replaced by the average of the central differences at time levels n and n+1, the discretized equation would be of the form: Note: The left side of eq. (8-4) is a central difference of step, i.e.,, which is

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9-2 Finite Difference Formulations (6)

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9-2 Finite Difference Formulations (7) (ii) The method may be thought of as the addition of two step computations as follows: Using the explicit method, while using the implicit method, Adding eqs. (8-5a) and (8-5b), we can get eq. (8-4). (iii) This implicit method is unconditionally stable and is of order, that is a second-order accurate scheme. Example

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9-3 Parabolic Equations in Two- Space Dimensions (1) Consider the model equation (1) FTCS (or Explicit) method: which is of order Stability analysis indicates that the method is stable for where If Δx=Δy, i.e., dx=dy=d, then

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9-3 Parabolic Equations in Two- Space Dimensions (2)

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9-3 Parabolic Equations in Two- Space Dimensions (3) (2) Implicit (BTCS) method: (i) Consider an implicit formulation for which the FDE is

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9-3 Parabolic Equations in Two- Space Dimensions (4) (ii) The 2-D FDEs in the ADI formulation are and

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9-3 Parabolic Equations in Two- Space Dimensions (5) This method is of order and is unconditionally stable. The above equations are written in the tridiagonal form as where

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9-3 Parabolic Equations in Two- Space Dimensions (6)

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CHAPTER 10 Stability Analysis

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10-1 Stability considerations (1) At a starting point for stability analysis, consider the simple explicit approximation to the heat equation This may be solved for u n j to yield

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10-1 Stability considerations (2) Let D be the exact solution of this equation(2), N the numerical solution of equation(1) and A the analytical solution of the PDE: Then,we may write Discretization error=A-D Round-off error=N-D

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10-1 Stability considerations (3) The equation of stability of a numerical method examine the error growth while computations are being performed. The equation of stability is usually answered by using a Fourier analysis. This method is also referred to as a von Neumann analysis.

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10-2 Fourier or von Neumann analysis (1) Consider eq.(1) and let εbe the round-off error. The numerical solution actually computed may be written N=D+ε----(3) N must satisfy eq.(1). Substituting eq.(3) into eq.(1), yields

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10-2 Fourier or von Neumann analysis (2) since the exact solution must satisfy the difference eq.(i.e. Eq(1)), therefore

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10-2 Fourier or von Neumann analysis (3) In this case, the exact solution D and the error εmust both satisfy the same difference equation. This means that the numerical error and exact numerical solution both posses the same growth property in time and either could be used to examine stability.

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10-2 Fourier or von Neumann analysis (4) Any perturbation of the input values at the n th time level will either be prevented from growing without bound for a stable system or will grow large for an unstable system

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10-2 Fourier or von Neumann analysis (5) Consider a distribution of error at any time in a mesh. We choose to view this distribution a time t=0 for convenience. This error distribution is shown ε (x,0) x

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10-2 Fourier or von Neumann analysis (6) We assume the error ε (x,t) can be written as a series of the form

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10-2 Fourier or von Neumann analysis (7) Since the difference equation is linear, superposition may be used, and we may examine the behavior of a single term of the series given in eq.(4). consider the term

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10-2 Fourier or von Neumann analysis (8) For an assessment of numerical stability, we are interested in the variation of with time. Therefore, we external eq.(2) by assuming the amplitude b m is a function of time. Moreover, it is reasonable to assume an exponential variation with time; error tend to grow or diminish exponentially with time. Therefore, we write

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10-2 Fourier or von Neumann analysis (8) where k m is real, but “a” may be complex. If eq(6) is substituted into eq(1), we obtain ……(7)

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10-2 Fourier or von Neumann analysis (9) If we divide by e at e ikmx and utilize the relation

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10-2 Fourier or von Neumann analysis (10) We can get:

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10-2 Fourier or von Neumann analysis (11) ∴ the error will not grow from one time step to the next, if

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