Conservative First Order Upwind Methods Nonlinear Case In the nonlinear case au, af( 0 t aC the flux becomes N4 UP 2(J+1 2j+ j+1 J+1- j+1-t fy+1≠ f()f+1= SMA-HPC⊙2003MT Hyperbolic Equations 10
Conservative Lax-Wendroff Methods FLw 1 j+-2 (n+月)-2△( For the linear equation 72+1 1 + +1-20 +a1) C=a△/△t SMA-HPC⊙2003MT Hyperbolic Equations 11
Conservative Beam-Warming Methods H=(-+2+8+1+8-1) 1)-0+a(+2-1+1-4-) (-+2+3-3+1)+819(4+2-+一+每1 + / For the linear equation 2+1 a-2(3的-4121+a2)+2(-21+2) >0 2+1 0-2(-307+4t7 2)+(a+2-2a3+1+)a< SMA-HPC⊙2003MT Hyperbolic Equations 12
Conservative Entropy Solutions Methods Do these schemes converge to the entropy satisfying solution? EXAMPLE: Consider a non-physical solution to Burgers' equation 1 >0 (,) 1a<0 e. in is either10r-1→f=的 SMA-HPC⊙2003MT Hyperbolic Equations 13
Conservative Entropy Solutions Methods Example First order upwind FH号=2(+1+月)-2+(2+1-) Since either ai +1 or ij+1-ij is zero Vj +=2的→FUP_pUP=0j →FO= j+3 7+1 The entropy-violating solution is preserved N5 SMA-HPC⊙2003MT Hyperbolic Equations 14