Computational Methods in Engineering: Introduction to Partial Differential Equations
Explore the basics of Parabolic Partial Differential Equations (PDE) and their semi-discretization techniques in computational engineering. Learn about common Finite Difference methods for solving various types of PDEs. Dive into the world of PDEs with a focus on diffusion, advection-diffusion, Laplace, and wave equations.
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ESO 208A: Computational Methods in Engineering Partial Differential Equations: Introduction, Parabolic Equation Department of Civil Engineering IIT Kanpur
Introduction A general 2ndOrder PDE: ????+ 2????+ ????+ ???+ ???+ ? ?,?,? = 0 ?2 ?? = 0: Parabolic PDE, e.g., Diffusion and Advection-Diffusion Equation ?2 ?? < 0: Elliptic PDE, e.g., Laplace Equation ?2 ?? > 0: Hyperbolic PDE, e.g., Wave equation We will learn a few Finite Difference methods for most common PDEs in Engineering Problems!
Parabolic PDE: semi-discretization ?? ??= ??2? ? 0,? = ?0; ? ?,? = ??; ? ?,0 = ? ? ??= ??2? ?? ??+ ??? ??2 ??2 O(h2) ??? ?? ??? ?? ??+1 2??+ ?? 1 ??2 ??+1 ?? 1 2?? = ?? O(h2) O(h2) ??+1 2??+ ?? 1 ??2 = ?? + ??
Parabolic PDE: semi-discretization ?? ??= ??2? ? 0,? = ?0; ? ?,? = ??; ? ?,0 = ? ? ; ? = 4 ??? ?? ??+1 2??+ ?? 1 ??2 ??2 = ?? ??1 ?? Now, ?0 = ? 0,? ?2 2?1+ ?0 ??2 = ?1 ??1 ?? ??2 ?? ??3 ?? 2?1 ??2 ?2 ??2 ?1 ??2 2?2 ??2 ?3 ??2 ?0?1 ??2 0 ???3 ??2 0 = ?0 ?1 ?2 ?3 ?2 ??2 2?3 ??2 = + 0
Parabolic PDE: semi-discretization ??= ??2? ?? ??+ ??? ? 0,? = ?0; ? ?,? = ??; ? ?,0 = ? ? ??? ?? ??+1 ?? 1 2?? ??+1 2??+ ?? 1 ??2 ??2 = ?? + ?? ??1 ?? ??2 ?? ??3 ?? 2?1 ??2 ?2 2??+ ?1 2??+ 2?2 ??2 ?3 2??+ ?1 ??2 ?1?0 2??+?0?1 0 ?3?? 0 ?1 ?2 ?3 ??2 ?2 ??2 ?2 2??+ 2?3 ??2 ?2 ??2 = + 2??+???3 ?3 ??2 ??2 0
Parabolic PDE: semi-discretization ?? ??= ??2? ? 0,? = ?0; ? ?,? = ??; ? ?,0 = ? ? ??= ??2? ?? ??+ ??? ??2 ??2 ? ?1 ? ?2 ? ?3 ?1 ?2 ?3 d ? dt= ? ? + ? ? = ? 0 =
Parabolic PDE: semi-discretization ? ?1 ? ?2 ? ?3 ?1 ?2 ?3 d ? dt= ? ? + ? ? = ? 0 = ?? ??= ? ?,? ??+?= ??+ ?? (Euler F) or ??+?= ??+ ??+? (Euler B) ??+1= ??+ ?? ? ?? ??+ ??+ 1 ? ??+1 ??+1+ ??+1 = 0: Euler Backward; = 1: Euler Forward = 1/2: Trapezoidal ? ?? 1 ? ??+1 ??+1 = ? + ????? ??+ ?? 1 ? ??+1+ ???
Parabolic PDE: full-discretization ?? ??= ??2? ? 0,? = ?0; ? ?,? = ??; ? ?,0 = ? ? ??2 O(h) ?+1 ?? ?? ???+1 ? ?? O(h2) O(h2) ? ?+ ?? 1 ? ?+1 2?? ?+1+ ?? 1 ??2 ?+1 2?? ??2 ?+1??+1 = ??? + 1 ? ??
Parabolic PDE: full-discretization ??= ??2? ?? ??+ ??? ? 0,? = ?0; ? ?,? = ??; ? ?,0 = ? ? ??2 O(h) ?+1 ?? ?? ? ?? O(h2) O(h2) ? ? ? ?+ ?? 1 ? ???+1 ?? 1 2?? ?+1??+1 ???+1 2?? ??2 ?+1??+1 O(h2) = ? ?? + ?? ?+1 ?? 1 2?? ?+1 ?+1 2?? ?+1+ ?? 1 ??2 ?+1 + 1 ? ?? + ??
Types of Boundary Conditions Dirichlet Condition (1st Type): Variable value is specified ? 0,? = ?0; ? ?,? = ?? Neumann Condition (2nd Type): Gradient is specified dj dx = c ( ) and/or L,t ( ) 0,t Robin Condition (3rd Type): A linear combination of the variable and gradient is specified at (0, t) and/or (L, t) adj dx+bf = c
Example ??= ??2? ?? ??+ ??? ??2 ? = 0.1; ? = 0.01;? 0,? = 0; ? 1,? = 0; ? ?,0 = 50sin?? Solve Using: (a) Euler Forward in time and Central Difference in space (EF-CD) (b) Euler Backward in time and Central Difference in space (EB-CD) (c) Crank Nicholson method (d) A 2nd order R-K method in time and Central Difference approximation in space. Use x = 0.25 and t = 0.5 3 2 0 1 4
Example ??= ??2? ?? ??+ ??? ? = 0.1; ? = 0.01;? 0,? = 0; ? 1,? = 0; ? ?,0 = 50sin?? ??2 3 2 0 1 4 O(h2) O(h2) +uTj+1-Tj-1 2Dx =aTj+1-2Tj+Tj-1 dTj dt Dx2 dTj dt Tj-1+ -2a Tj+ - Tj+1 2Dx+a 2Dx+a u u = Dx2 Dx2 Dx2 dTj dt = 0.36Tj-1-0.32Tj-0.04Tj+1
Example ??= ??2? ?? ??+ ??? ? = 0.1; ? = 0.01;? 0,? = 0; ? 1,? = 0; ? ?,0 = 50sin?? ??2 3 2 0 1 4 dT dt dTj dt = AT = 0.36Tj-1-0.32Tj-0.04Tj+1 T1 T2 T3 0 T1 T2 T2 -0.32 0.36 0 -0.04 -0.32 0.36 0 35.3553 50.0 35.3553 T = A = -0.04 -0.32 T0= = 0 0
Example 0 -0.32 0.36 0 -0.04 -0.32 0.36 T1 T2 T2 0 T1 T2 T3 35.3553 50.0 35.3553 dT dt A = -0.04 -0.32 = AT T0= = 0 T = 0 Euler Forward: Tn+1= I +DtA Tn 0.5 T1 T2 T2 -0.02 0.84 0.18 0.84 0.18 0 0 35.3553 50.0 35.3553 28.6985 47.6568 38.6985 T0.5= = -0.02 0.84 = 0.5 0.5
Example 0 -0.32 0.36 0 -0.04 -0.32 0.36 T1 T2 T2 0 T1 T2 T3 35.3553 50.0 35.3553 dT dt A = -0.04 -0.32 = AT T0= = 0 T = 0 Euler Backward: Tn+1= Tn I -DtA 0.5 T1 T2 T2 1.16 -0.18 0 0.02 1.16 -0.18 1.16 0 35.3553 50.0 35.3553 = 0.5 0.02 0.5 0.5 T1 T2 T2 29.6674 47.0556 37.7804 T0.5= = 0.5 0.5
Example 0 -0.32 0.36 0 -0.04 -0.32 0.36 T1 T2 T2 0 T1 T2 T3 35.3553 50.0 35.3553 dT dt A = -0.04 -0.32 = AT T0= = 0 T = 0 Crank-Nicholson: Tn+1= I +0.5DtA Tn I -0.5DtA 0.5 T1 T2 T2 -0.01 0.92 0.09 1.08 -0.09 0 0.01 1.08 -0.09 1.08 0 0.92 0.09 0 0 35.3553 50.0 35.3553 32.0269 48.8284 37.0269 = -0.01 0.92 = 0.5 0.02 0.5 0.5 T1 T2 T2 29.2166 47.2923 38.2252 T0.5= = 0.5 0.5
Example 0 -0.32 0.36 0 -0.04 -0.32 0.36 T1 T2 T2 0 T1 T2 T3 35.3553 50.0 35.3553 dT dt A = -0.04 -0.32 = AT T0= = 0 T = 0 2nd Order Runge-Kutta (Heun s Predictor-Corrector Form): n+1= I +DtA Tn Tp This is same as Euler-Forward. EF solution is the Predictor. = Tn+Dt n+1= Tn+Dt n+1+ATn n+1+Tn Tc 2ATp 2A Tp 0.5 T1c T2c T2c -0.08 0.09 0 -0.01 -0.08 0.09 35.3553 50.0 35.3553 0 28.6985 47.6568 38.6985 35.3553 50.0 35.3553 29.2544 47.2118 38.2201 0.5= = + -0.01 -0.08 + = 0.5 Tc 0.5
Convergence What about Consistency, Stability, Convergence? How does one choose x and t? Are they interdependent? Diffusion Equation: ?? ?? ???+1 2?? ??2 ?+1 ?? ? ? ?+ ?? 1 ? ?+1 2?? ?+1+ ?? 1 ??2 ?+1 ?+1??+1 = ??? + 1 ? ?? fi+1 fi fi fi-1 n+1Dt Dx2 n+1Dt Dx2 n+1Dt Dx2 - 1- m ( )ai ( )ai ( )ai n+1+ 1+2 1- m nDt Dx2 n+1+ - 1- m n+1 fi+1 fi-1 nDt Dx2 nDt Dx2 = mai n+ 1-2mai n+ mai n
Convergence Advection-Diffusion Equation: ?+1 ?? ?? = ? ?? ? ?? ? ? ? ?+ ?? 1 ? ???+1 ?? 1 2?? ?+1??+1 ???+1 2?? ??2 ?+1??+1 + ?? ?+1 ?? 1 2?? ?+1 ?+1 2?? ?+1+ ?? 1 ??2 ?+1 + 1 ? ?? + ?? fi+1 fi-1 fi n+1Dt 2Dx-ai n+1Dt Dx2 n+1Dt Dx2 n+1Dt 2Dx-ai n+1Dt Dx2 1- m ( ) ui ( )ai n+1+ 1- m ( fi ) -ui n+1+ 1+2 1- m n+1 fi+1 fi-1 nDt 2Dx+ai nDt Dx2 nDt Dx2 n Dt 2Dx+ai nDt Dx2 = m -ui n+ 1-2mai n+ m ui n Groups ? ? ? ?2 govern the equations. ? and ?
Convergence Peclet Number: ??=?? ?=?? ? Grid Peclet Number: ??=? ? =? ? ? ? CFL (Courant-Friedrich-Lewy) Number: ? = ? ? ? Therefore, ? ?? ? ?2 = ?
Convergence If u and ? are constants (not function of x): fi+1 fi-1 fi ) uDt 2Dx-aDt )aDt ) -uDt 2Dx-aDt 1- m ( = m -uDt ( n+1+ 1- m ( fi n+1+ 1+2 1- m 2Dx+aDt Dx2 n+1 Dx2 Dx2 Dx2 fi-1 fi+1 n+ 1-2maDt n+ m uDt 2Dx+aDt n Dx2 Dx2 C 2-C )C Pg ) -C 2-C 1- m ( ) ( 1- m ( n+1+ 1+2 1- m n+1+ n+1 fi+1 fi fi-1 Pg Pg = m -C 2+C n+ 1-2mC C 2+C fi+1 fi n+ m fi-1 n Pg Pg Pg Then the solutions depend on these two dimensionless groups or numbers. Therefore, stability and convergence will also depend on these two!
