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Computational Fluid Dynamics I PIW Numerical Methods for Parabolic Equations Instructor: Hong G. Im University of Michigan Fall 2005
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Computational Fluid Dynamics I PIW One-Dimensional Problems Explicit, implicit, Crank-Nicolson Accuracy, stability Various schemes Keller Box method and block tridiagonal system Multi-Dimensional Problems Alternating Direction Implicit (ADI) Approximate Factorization of Crank-Nicolson Outline Solution Methods for Parabolic Equations
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Computational Fluid Dynamics I PIW Numerical Methods for One-Dimensional Heat Equations
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Computational Fluid Dynamics I PIW which is a parabolic equation requiring In this lecture, we consider a model equation Initial Condition Boundary Condition (Dirichlet) Boundary Condition (Neumann) or and
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Computational Fluid Dynamics I PIW Explicit: FTCS j 1 j j+1 n n+1 Explicit Method: FTCS - 1
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Computational Fluid Dynamics I PIW Modified Equation where - Accuracy - If then - No odd derivatives; dissipative Explicit Method: FTCS - 2
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Computational Fluid Dynamics I PIW Stability: von Neumann Analysis (Recall Lecture 2, p. 60) Fourier Condition Explicit Method: FTCS - 3
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Computational Fluid Dynamics I PIW
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PIW
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PIW Domain of Dependence for Explicit Scheme BC x t Initial Data h P tt Boundary effect is not felt at P for many time steps This may result in unphysical solution behavior Explicit Method: FTCS - 4
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Computational Fluid Dynamics I PIW Implicit Method: Laasonen (1949) j 1 j j+1 n n+1 Tri-diagonal matrix system Implicit Method - 1
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Computational Fluid Dynamics I PIW - The + sign suggests that implicit method may be less accurate than a carefully implemented explicit method. Modified Equation Implicit Method - 2 Amplification Factor (von Neumann analysis) Unconditionally stable
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Computational Fluid Dynamics I PIW Crank-Nicolson Method (1947) j 1 j j+1 n n+1 Tri-diagonal matrix system Crank-Nicolson - 1
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Computational Fluid Dynamics I PIW - Second-order accuracy Modified Equation Crank-Nicolson - 2 Amplification Factor (von Neumann analysis) Unconditionally stable
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Computational Fluid Dynamics I PIW Generalization j 1 j j+1 n n+1 Combined Method A - 1
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Computational Fluid Dynamics I PIW Other Special Cases (for further accuracy improvement): Combined Method A - 2 (a) If (b) If and Modified Equation
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Computational Fluid Dynamics I PIW unconditionally stable Combined Method A - 3 Stability Property stable only if
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Computational Fluid Dynamics I PIW Generalized Three-Time-Level Implicit Scheme: Richtmyer and Morton (1967) j 1 j j+1 n n+1 Combined Method B - 1 n1n1
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Computational Fluid Dynamics I PIW Modified Equation: Combined Method B - 2 Special Cases: (a) If (b) If
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Computational Fluid Dynamics I PIW Richardson Method: A Case of Failure Richardson Method Similar to Leapfrog but unconditionally unstable! j 1 j j+1 n n+1 n1n1
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Computational Fluid Dynamics I PIW The Richardson method can be made stable by splitting by time average j 1 j j+1 n n+1 DuFort-Frankel - 1 n1n1
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Computational Fluid Dynamics I PIW Modified Equation (Recall Homework #1) DuFort-Frankel - 2 Amplification factor Unconditionally stable Conditionally consistent
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Computational Fluid Dynamics I PIW FTCS Stable for BTCS Unconditionally Stable Crank-Nicolson Unconditionally Stable Richardson Unconditionally Unstable Parabolic Equation - Summary
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Computational Fluid Dynamics I PIW DuFort-Frankel Unconditionally Stable, Conditionally Consistent 3-Level Implicit Unconditionally Stable Parabolic Equation - Summary
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Computational Fluid Dynamics I PIW The Keller Box Method (1970) Keller Box - 1 - Implicit with Basic Concept: Define yielding
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Computational Fluid Dynamics I PIW In discretized form Keller Box - 2 j 1 j n+1 n t n+1 hjhj j 1 j n+1 n where
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Computational Fluid Dynamics I PIW Substituting Keller Box - 3 or
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Computational Fluid Dynamics I PIW Adding boundary conditions, e.g. Keller Box - 4
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Computational Fluid Dynamics I PIW In Matrix Form Keller Box - 5 D1D1 A1A1 A2A2 D2D2 B2B2 B3B3 D3D3 A3A3 DJDJ BJBJ Block Tridiagonal Matrix Appendix, Linpack
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Computational Fluid Dynamics I PIW Modified Keller Box Method – Express in terms of Keller Box - 6 Starting with original Keller box method: Eliminating from (b) using (a) (a) (b)
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Computational Fluid Dynamics I PIW Further elimination of yields (Tannehill, p. 136) Keller Box - 7 Tridiagonal Matrix Thomas Algorithm
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Computational Fluid Dynamics I PIW Notes on Keller Box Method Keller Box - 8 1.Second-order accurate in time and space 2.Accuracy is preserved for nonuniform grids 3.More operations per timestep compared to Crank-Nicolson
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Computational Fluid Dynamics I PIW Numerical Methods for Multi-Dimensional Heat Equations
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Computational Fluid Dynamics I PIW Applying forward Euler scheme: Consider a 2-D heat equation If Explicit Method - 1
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Computational Fluid Dynamics I PIW In matrix form Explicit Method - 2
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Computational Fluid Dynamics I PIW Explicit Method - 3 Von Neumann Analysis Worst case
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Computational Fluid Dynamics I PIW Explicit method for multi-dimensional heat equation is not desirable. Explicit Method - 4 Stability Condition for Heat Equation (2-D) (3-D) (1-D)
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Computational Fluid Dynamics I PIW Too expensive!! Crank-Nicolson Crank-Nicolson Method for 2-D Heat Equation If
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Computational Fluid Dynamics I PIW Fractional Step: ADI - 1 A Clever Remedy – Alternating Direction Implicit (ADI) Step 1: Step 2: Combining the two becomes equivalent to:
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Computational Fluid Dynamics I PIW ADI - 2 Computational Molecules for ADI Method n n+1/2 n+1 i i+1 i1i1 j+1 j1j1 j
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Computational Fluid Dynamics I PIW Stability Analysis: ADI - 3 ADI Method is accurate Similarly,
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Computational Fluid Dynamics I PIW ADI - 4 Combining Unconditionally stable! Unfortunately, a 3-D version does not have the same desirable stability properties. Conditionally stable and
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Computational Fluid Dynamics I PIW ADI - 5 Advantages: 1.Stability limits of 1-D case apply. 2.Different t can be used in x and y directions. 1. Cannot be directly extended to 3-D problems. Approximate factorization Limitations:
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Computational Fluid Dynamics I PIW Approximate Factorization - 1 The Crank-Nicolson for heat equation becomes Define which can be rewritten as
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Computational Fluid Dynamics I PIW Approximate Factorization - 2 Factoring each side or
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Computational Fluid Dynamics I PIW Approximate Factorization - 3 Final factored form of the discrete equation at an accuracy of Note that the ADI method can be written as which is equivalent to factorized Crank-Nicolson
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Computational Fluid Dynamics I PIW Approximate Factorization - 4 Two-step algorithm can be written into two steps: each of which can be solved by TDMA (Thomas algorithm). (Step 2) (Step 1)
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Computational Fluid Dynamics I PIW Approximate Factorization - 5 Note: Boundary condition for is needed at This can be determined from For example, at Similarly, at
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Computational Fluid Dynamics I PIW Approximate Factorization - 6 Generalized 3-D Algorithm: unconditionally stable, where (Step 2) (Step 1) (Step 3)
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