Scientific Computing I

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1 Scientific Computing I Session 11: Basics of Partial Differential Equations Tobias Weinzierl Winter 2010/2011 Session 11: Basics of Partial Differential Equations, Winter 2010/2011 1

3 Shortcomings of ODEs Sometimes/often, the behaviour of a point in space does not depend solely on its history but also on its neighbours. (C) 2010 Tobias Köppl Session 11: Basics of Partial Differential Equations, Winter 2010/2011 3

4 PDE Operators: Time Derivative Operator does not depend on its neighbours. t t u = f on Ω R d can be seen as limit of set of independent ODEs. This is defined component-wise for any u : Ω R k. Session 11: Basics of Partial Differential Equations, Winter 2010/2011 4

5 PDE Operators: The Heat Equation Let u : Ω R. Heat flow is proportional to temperature gradient q (left) ij kh y T ij T i 1,j h x Study arbitrary (small) control volume D Change of heat energy (per time) is a result of Transfer of heat energy across D s surface, Heat sources and drains in D (external influences) Resulting integral equation: t D ρct dv = D q dv + D k T n ds density ρ, specific heat c, and heat conductivity k are material parameters Heat sources and drains are modelled in term q Session 11: Basics of Partial Differential Equations, Winter 2010/2011 5

6 PDE Operators: The Heat Equation Part II According to theorem of Gauß: k T n ds = k T dv D D Leads to integral equation for any domain D: ρct t q k T dv = 0 D Hhence, the integrand has to be identically 0: T t = κ T + q ρc, κ := k ρc κ > 0 is called the thermal diffusion coefficient (since the Laplace operator models a (heat) diffusion process) Session 11: Basics of Partial Differential Equations, Winter 2010/2011 6

7 PDE Operators: The Heat Equation Part III Different scenarios: Vanishing external influence, q = 0: T t = κ T alternate notation T t ( 2 ) T = κ x T y T z 2 Equilibrium solution, T t = 0: 0 = κ T + q ρc T = f Poisson s Equation For the instationary/transient case, it is a combination of the Poisson operator (diffusion operator) and the time derivative. Session 11: Basics of Partial Differential Equations, Winter 2010/2011 7

8 PDE Operators: Incompressibility Study an incompressible fluid, and, again, study a control volume D. Let u : Ω R d be a vector function. For our control volume: mass change = 1 D D (u, n)ds(x) Mass/number of molecules entering D has to be equal to the mass leaving D: 1 lim (u, n)ds(x) = x u x + y u y + z u z = div u h 0 D D Divergence operator. Session 11: Basics of Partial Differential Equations, Winter 2010/2011 8

9 PDE Operators: Internal Friction Study a fluid running from left to right. Let u : Ω R d be a vector function. Consider one particle/molecule. If upper neighbour particle is faster, it speeds up. If upper neighbour particle is slower, it slows down. Same reasoning for lower particle. The difference and the distance matter: influcence up 1 h (u upper u me ) y u The difference and the distance of both neighbouring points matter: effect = 1 h (influcence up influcence down ) y y u Internal friction is again a Laplace operator: t u u = 0 Session 11: Basics of Partial Differential Equations, Winter 2010/2011 9

10 PDE Operators: Transport Equation Study a fluid running from left to right that has a temperature. Let u : Ω R d be a vector function, and T : Ω R is the fluid s temperature. Consider one particle/molecule. It is transported by u, and, consequently, its temperature is transported by u. However, we won t recognise this (for one fixed point), if the temperature is the same around this fixed point. We only recognise it, if the temperature differs in the surrounding areas and we see a shift of the temperature field due to the fluid s movement. Operator: u T = ( u 1 u 2 u 3, x T y T z T ) = u 1 x T + u 2 y T + u 3 z T Session 11: Basics of Partial Differential Equations, Winter 2010/

11 PDE Operators: Convection Operator Study a fluid running from left to right that has a temperature. Let u : Ω R d be a vector function. Also the particles are dragged/carried by their neighbours. Instead of the temperature, plug in the velocity field itself, and make the operator act component-wise: u u = u 1 u u 2 u u 3 u Convection operator (reason for turbulence in fluids) couples different velocity components. Session 11: Basics of Partial Differential Equations, Winter 2010/

12 Writing Down PDE Operators: Different Styles Idea 1: Write one symbol per operator:, div, Idea 2: Write operators as vectors/matrices of derivatives: x u 1 u = y z u div u = ( x y z ) u 2 u 3 We see directly: u = ( x y z ) x y z u div = T = div = T Idea 3 (Einstein): Write operators as sums with free arguments u i = j j u i Session 11: Basics of Partial Differential Equations, Winter 2010/

Part I. Discrete Models. Part I: Discrete Models. Scientific Computing I. Motivation: Heat Transfer. A Wiremesh Model (2) A Wiremesh Model

Part I. Discrete Models. Part I: Discrete Models. Scientific Computing I. Motivation: Heat Transfer. A Wiremesh Model (2) A Wiremesh Model Part I: iscrete Models Scientific Computing I Module 5: Heat Transfer iscrete and Continuous Models Tobias Neckel Winter 04/05 Motivation: Heat Transfer Wiremesh Model A Finite Volume Model Time ependent