Numerical Methods for Hyperbolic Conservation Laws: A Mathematical Theory, Study notes of Mathematics

Download Numerical Methods for Hyperbolic Conservation Laws: A Mathematical Theory and more Study notes Mathematics in PDF only on Docsity! Numerical Methods for Hyperbolic Conservation Laws Mathematical Theory for Systems Dr. Aamer Haque http://math.iit.edu/~ahaque6 ahaque7@iit.edu Illinois Institute of Technology June 9, 2009 Outline  Review  Linear Systems  1D Linearized Flow  Nonlinear Systems  1D Euler Equations Hyperbolic Conservation Laws A system of conservation laws is hyperbolic if the Jacobian matrix has real eigenvalues and a corresponding set of linearly independent right eigenvectors The system is strictly hyperbolic if the eigenvalues are distinct The system is linear if the Jacobian matrix does not depend on The matrix is diagonalizable if it can be expressed as 1 ,2 , ,m A k 1 , k 2 , , k m m A U A=K K−1 =K−1 A K =[1 ⋯ 0⋮ ⋱ ⋮0 ⋯ m] K=[k 1 ⋯ k m] A Characteristic Variables W=K−1U We will focus on the linear case with constant Jacobian matrix For strictly hyperbolic systems, this matrix is diagonalizable We can transform the conserved quantities to characteristic variables The PDE for the system can be written in characteristic form This is a decoupled system of scalar advection equations The solution is A ∂W ∂ t  ∂W ∂ x =0 U=KW ∂w i ∂ t i ∂w i ∂ x =0 w i x , t =w i 0 x−i t  W=[w1 ⋯ wm ] T W 0=K−1U 0 U x , t =∑i=1 m wi 0x−i t k i Linear Systems - Riemann Problem U 0={U L x0U R x0} Initial Conditions Solution w i x , t =w i 0 x−i t ={i xi ti xi t} U x , t =∑i=I1 m i k i i∑i=1 I i k i U L=∑i=1 m i k i U R=∑i=1 m i k i w i 0x={i x0i x0} 1D Linearized Flow - Characteristics x x  , u , t t=0 t=t1 L u* R L R * uL=0 uR=0 Characteristic Fields The i-th characteristic field is genuinely nonlinear if The i-th characteristic field is linearly degenerate if ∇i U ⋅k i U ≠0,∀U∈ℝm ∇i U ⋅k i U =0,∀U∈ℝm ∇i U =[ ∂i∂u1 ⋯ ∂i∂um ] T Nonlinear Systems – Shock Wave x t U L U R F U R−F U L=S i U R−U L S i i U LS ii U R Genuinely Nonlinear Field Rankine-Hugoniot Jump Condition Entropy Condition 1D Euler Equations ∂U ∂ t  ∂ ∂ x F U =0 U=[ u E] F U =[ uu2 p E pu] ∂U ∂ t AU  ∂U ∂ x =0 k 1=[ 1u−cH−uc] 1=u−c k 2=[ 1uu2/2] 2=u k 3=[ 1ucHuc] 3=uc H=h1 2 u2 h=e p Conservation Form Conserved Variables Flux Function Quasi-linear Form Enthalpy Eigenvalues and Eigenvectors Quasi-linear Form Primitive Variables Matrix Eigenvalues and Eigenvectors 1D Euler Equations - Primitive Form ∂W ∂ t AU  ∂W ∂ x =0 W=[up] AU =[u  00 u 1/0 c2 u ] k 1=[ 1−c /c2 ] 1=u−c k 2=[100] 2=u k 3=[ 1c /c2 ] 3=uc l1=[0 1 −1c ] l3=[0 1 1c ]l2=[1 0 −1c2 ] Quasi-linear Form Primitive Variables Matrix Eigenvalues and Eigenvectors 1D Euler Equations - Entropy Form ∂W ∂ t AU  ∂W ∂ x =0 W=[us ] AU =[ u  0c2/ u 1 ∂ p∂ s0 0 u ] k 1=[ 1−c /0 ] 1=u−c k 2=[−∂ p∂ s0c2 ] 2=u k 3=[ 1c /0 ] 3=uc 1D Euler Equations Contact Discontinuity x t U L U R S i Linearly Degenerate Field Riemann Invariants Constants Across Wave 2 d  1 = d u u = d  E  u2/2 p=const u=const Characteristic Equations can be derived using along 1D Euler Equations - Characteristics lidW=0 dx dt =1=u−c dx dt =2=u dx dt =3=uc d − 1 c2 dp=0 du− 1 c dp=0 du 1 c dp=0 J +=u∫ c d =const J -=u−∫ c d =const s=const dx dt =i Characteristic Nets x t t=0 J±=u± 2 −1 c J +J -