Solving Partial Differential Equations Numerically. Miklós Bergou with: Gary Miller, David Cardoze, Todd Phillips, Mark Olah

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1 Solving Partial Dierential Equations Numerically Miklós Bergou with: Gary Miller, David Cardoze, Todd Phillips, Mark Olah

2 Overview What are partial dierential equations? How do we solve them? (Example) Numerical integration Doing this quickly

3 Partial Dierential Equations Equation involving unctions and their partial derivatives Example: Wave Equation We wish to know ψ, which is unction o many variables Typically, no analytical solution possible t v z y x + + ψ ψ ψ ψ

4 Problem Domain Want to solve problem or speciic domain : bounded open domain in space R n Γ: boundary o I domain -D, we have ollowing: Γ

5 Navier-Stokes Equation Model o incompressible luid low Governed by equation: u + t u 0 ( u ) u 0 u P u ν u + a in (0,T] ρ in (0,T] on Γ (0,T] 0 ( x) u at t 0 u: luid velocity P: pressure ρ: mass density ν: dynamic viscosity a: acceleration due to external orce

6 Simpliying Navier-Stokes: Just Stokes Assume steady low: ( u ) u u 0 u u 0 u ν 0 0 P u ρ Neglect convection: P ν u ρ + a + a in in on Γ in in on Γ

7 Solving Stokes Equation ν u u 0 u 0 on Γ Space o possible unctions V that could be solutions to velocity u Choose any v V, multiply both sides o Stokes Equation and integrate, resulting in: ν udx v in in v dx

8 Solving Stokes Equation Apply Green s Theorem and boundary conditions to obtain: ν u vdx For notational convenience, this is written as: u, v (, v) vdx

9 Discretize the domain Create mesh by partitioning domain into inite elements (curved Bezier triangles) Create subspace V h o V o piecewise polynomial unctions with basis unction deined at each node ϕ v ( x) set o basis unctions ( x) v ( ) ( ) h x ηiϕi x h V h M i

10 Linear Equations Since V h V, u h V so we can say that in order or u h to be a solution to Stokes equation, we need u h, v (, v) v Vh In particular, must be true or basis unctions, so writing u h in terms o basis unctions, we have: M i η ( ) ϕ j M ϕ, ϕ,,..., i i j j

11 Solution to Problem Aη b A η b ij ϕ, ϕ i [ η,..., η ] [(, ϕ ),...,(, ϕ )] T Knowing coeicients η i means we know solution u h This is a system o M linear equations with M unknowns, can be solved with various numerical methods j M T M

12 Integration Computing stiness matrix and load vector requires lots o integrals ϕ i, ϕ j, (,ϕ i ) These are done numerically via quadratures: 0 y 0 ( x) dx 0 n i w ( x, y) dxdy ( x, y ) Choose Gauss points and weights to give high order accuracy i ( x ) n i i w i i i

13 Full Navier-Stokes Equation Requires our types o integrals and g are basis unctions on curved triangles Map rom K simplex to curved triangles: F : K Bezier β α α g g g g

14 Mapping K to Bezier Mapping requires 6 control points, deined by b i s are Bezier basis unction (polynomials) Jacobian is deined in standard way 6 i c i b i F y F x F y F x F J y y x x

15 Integrals on K β α α g g g g det det det det K K K T T K J J J J J J J J J β α α β ψ α ψ α ψ φ ψ φ ψ φ

16 Need or speed Each type o integral is done on each triangle, or each combination o basis unctions Ater each time step, mesh moves (Lagrangian), so integrals need to be done or each timestep Speed and accuracy are required

17 Speedup: Cacheing Cache K basis unctions, so only Jacobian needs to be evaluated or each element Basis unctions cached at Gauss points, so unctions are essentially just arrays o values Cache Jacobian or each element as well, since it is reused in every integral Large speedup versus recomputation each time

18 Idea: Expand integrals Jacobian determinant can be written in terms o Bezier basis unctions with coeicients given by control points 6 g φ ψ det J n i K i K φ ψ b Integrals in this sum no longer depend on control points Precompute integrals, compute coeicients only i

19 Reusing old values Ater each time step, large portions o mesh unchanged I Jacobian o element is close enough to old value, reuse old integrals Speedup depends on measure o close enough and how much mesh changes

20 Example movie

21 Questions?

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