# Interdisciplinary Applied Mathematics

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tained, we convert them back to the conservative variables and update the elemental interface values as follows:

P

pu

pv

E

R4 R3

Ri +

c

7-1

nx + R2ny

R4 R3 -x-ny

R2nx

^ R3 + R4 ^

+ (Puo),

I + (pvo ),

— ^pu0 • Uo + (pu)ua + (pv)va.

The above elemental interface treatment is one-dimensional and can be used in multiple dimensions by directional splitting. However, it cannot be used directly at element corners. To this end, we apply the treatment suggested in (Kopriva, 1991). The problem is divided into two one-dimensional problems. The corresponding Riemann invariants are

+ 2

= и H—cs; R

Y — 1

S+ = v H——rcs’, S’

Y-1

2

U-Cs,

Y-1

v —

2

Y-1

Cs.

We assume locally isentropic flow in the neighborhood of the corner and obtain the    entropy    value    of    the    corner from    the    element    that    the flow is

leaving. We also define a domain of influence, and choose the calculated values of Riemann invariants from the corresponding elements, which lie in the domain of dependence. Figure 14.3 shows the domain of influence at the    corners    of    four    elements.    For    this specific    example,    since    the    flow is

leaving element 1, we get the entropy and Riemann invariants of R+ and S + from element 1. The Riemann invariants R and S are obtained from elements 2 and 4, respectively. The flow variables are calculated as follows:

u=l-{R+ + R-)]    v=l-{S++S-),

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