# Interdisciplinary Applied Mathematics

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where Ai are the eigenvalues of A(Wo), and Ri(W,Wo) are the characteristic variables. For Euler equations the eigenvalues and the corresponding characteristic variables are

Ai =    A2    = Uo    • n,    A3    = Uo • n —    cs,    A4    = Uo • n +    cs,

Ri

R2

R3

P

Cs — [ pvLa • uG — m • uG + E I ,

R4

7 — 1 V2

(pu — puo)ny — (pv — pvo)nx,

-(m — puG) n + e ^ Qpu0 UG — m uG + E (m — puG) • n + f 7 e1 j ( ^puG • uG — m • ua + E ) .

Here cs is the local speed of sound, m is the momentum flux vector (m = (pu,pv)), E is the energy (E = pT + 0.5p(u2 + v2)), uo is the Roe-averaged velocity vector, and n is the surface normal, with nx and ny the x and components of the surface normal, respectively. The subscript o corresponds to the Roe-averaged states obtained by averaging two states (denoted by left, L, and right, R) as follows:

uo

ydpL’UL + yff>RUR _    _ л/~PlVL + ^J~PrVr

J~Pl +1/№    ’    ° a/Pl + a/Pr

The characteristic treatment is performed as follows: Each elemental interface is treated by analyzing the sign of the eigenvalues, Ai. We define the right (R) and the left (L) states at an interface depending on the local flow direction.    We    linearize    the    equations    around    the    Roe-averaged    states    ob

tained from the right and left values. If the sign of the eigenvalue is positive, we choose the characteristic variables from the left calculated state, and if the sign of the eigenvalue is negative, we choose the characteristic variables from the    right    calculated    state.    Once    the    characteristic    variables    are    ob

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