Quest for Osher-Type Riemann Solver

Transcription

1 p./7 Quest for Osher-Type Riemann Solver for Ideal MHD Equations Project Proposal EMRAS 2 Hervé Guillard, arry Koren, oniface Nkonga IRM, July 9 August 27, 2

2 p.2/7 Wine guide to Riemann solvers Godunov: Produced from vines of great antiquity. Full-bodied vintage to satisfy the most demanding palate. Modern viniculture is able to deliver most of its features at substantially lower cost though.

3 p.2/7 Wine guide to Riemann solvers Godunov: Produced from vines of great antiquity. Full-bodied vintage to satisfy the most demanding palate. Modern viniculture is able to deliver most of its features at substantially lower cost though. Roe: Moderately robust and finely flavored. Its variable density makes it unique. e warned, though, that its method of manufacture may cause very local headache in some states.

4 p.2/7 Wine guide to Riemann solvers Godunov: Produced from vines of great antiquity. Full-bodied vintage to satisfy the most demanding palate. Modern viniculture is able to deliver most of its features at substantially lower cost though. Roe: Moderately robust and finely flavored. Its variable density makes it unique. e warned, though, that its method of manufacture may cause very local headache in some states. Harten Lax van Leer: Robust wine that may be served on most occasions, but the omission of certain traditional ingredients leads to cloudiness and a sticky aftertaste.

5 p.2/7 Wine guide to Riemann solvers Godunov: Produced from vines of great antiquity. Full-bodied vintage to satisfy the most demanding palate. Modern viniculture is able to deliver most of its features at substantially lower cost though. Roe: Moderately robust and finely flavored. Its variable density makes it unique. e warned, though, that its method of manufacture may cause very local headache in some states. Harten Lax van Leer: Robust wine that may be served on most occasions, but the omission of certain traditional ingredients leads to cloudiness and a sticky aftertaste. Osher: Robust, smooth top-quality wine with a high proof content. From the best alifornian winery. Not a cheap wine to speed the gaiety. No artificial additives have been used. In my opinion the best buy!

6 p.3/7 Properties Osher scheme Strengths losest similarity to Godunov onsistent boundary-condition treatment onsistent source-term treatment ontinuous differentiability

7 p.3/7 Properties Osher scheme Strengths losest similarity to Godunov onsistent boundary-condition treatment onsistent source-term treatment ontinuous differentiability hallenges onstruction omputational intensity

8 p.4/7 Osher scheme in a nutshell onsider q such that: t + f(q) x = and suppose f + (q) and f (q) exist f(q) = f + (q) + f (q). Then, a natural approximate Riemann solver is: which can also be written as: F (q l, q r ) = f + (q l ) + f (q r ), F (q l, q r ) = f(q l ) f (q l ) + f (q r ) = f(q l ) + or : F (q l, q r ) = f(q r ) f + (q r ) + f + (q l ) = f(q r ) q r q l q r df dq dq, q l df + dq dq.

9 p.4/7 Osher scheme in a nutshell onsider q such that: t + f(q) x = and suppose f + (q) and f (q) exist f(q) = f + (q) + f (q). Then, a natural approximate Riemann solver is: which can also be written as: F (q l, q r ) = f + (q l ) + f (q r ), F (q l, q r ) = f(q l ) f (q l ) + f (q r ) = f(q l ) + or : F (q l, q r ) = f(q r ) f + (q r ) + f + (q l ) = f(q r ) q r q l q r Osher s elegant choice: integration along eigenvectors. df dq dq, q l df + dq dq.

10 Task onstruct Osher scheme for D ideal MHD equations, = s(q), with Ken Powell s source terms: q t + f(q) x q ρ ρu ρu 2 ρu e, f(q) = ρu ρu 2 + p ρu u 2 2 ρu u u u 2 3 u u 3 u `e + p + 2 u, s(q) = div 2 3 u u 2 u 3 u. A K.G. POWELL, An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension), WI Report NM-R947, Amsterdam (994). p.5/7

11 Task 2 onstruct Osher scheme for D ideal MHD equations, q t + f(q) x = s(q), with divergence cleaning terms: q ρ ρu ρu 2 ρu e ψ A, f(q) ρu ρu 2 + p ρu u 2 2 ρu u 3 3 ψ 2 u u 2 3 u u 3 u `e + p + 2 u c 2 h A, s(q) div div 2 div 3 ψ x c2 h c 2 p ψ A. A. DEDNER ET AL., Hyperbolic divergence cleaning for the MHD equations, J. omput. Phys., 75, (22). p.6/7

12 Success! p.7/7