Implement regularizations for Euler eq

Implements the Navier-Stokes and general regularization from J.-L. Guermond and B. Popov, “Viscous regularization of the euler equations and entropy principles,” SIAM Journal on Applied Mathematics, vol. 74, no. 2, pp. 284–305, 2014.

src/EulerEq/equation.jl

@@ -448,3 +448,71 @@ end
 
   @returns nflx_rho, nflx_qx, nflx_qy, nflx_E
 end
+
+
+# TODO Rename all methods accordingly
+const mono_ldg_nflux   = ldg_nflux
+const mono_ldg_bdryllf = ldg_bdryllf
+const mono_av_flux     = av_flux
+const mono_av_nflux    = av_nflux
+const mono_bdryllf     = bdryllf
+
+
+#######################################################################
+#                    Navier-Stokes regularization                     #
+#######################################################################
+
+
+@with_signature [legacy=false] function navierstokes_u(equation::EulerEquation)
+  @accepts rho, q
+  u = q/rho
+  @returns u
+end
+
+@with_signature [legacy=false] function navierstokes_ldg_nflux(eq::EulerEquation)
+  @accepts        u
+  @accepts [bdry] bdry_u
+  @accepts [bdry] nx
+
+  nflx_u = 0.5 * nx * (u + bdry_u)
+
+  @returns [bdry] nflx_u
+end
+@with_signature [legacy=false] function navierstokes_ldg_bdryllf(eq::EulerEquation)
+  @accepts        u
+  @accepts [bdry] nx
+
+  # set exterior state to 0 according to stability analysis
+  # see dg-notes/tex/evm.pdf for reasoning
+  nflx_u = nx * u
+
+  @returns [bdry] nflx_u
+end
+
+@with_signature [legacy=false] function navierstokes_av_flux(eq::EulerEquation)
+  @accepts flx_q, flx_E, ldg_u, u, smoothed_mu
+
+  # μ = λ, κ = 0
+  G = 4 * smoothed_mu * ldg_u + smoothed_mu * ldg_u
+  flx_q   += G
+  flx_E   += G * u
+
+  @returns flx_q, flx_E
+end
+@with_signature [legacy=false] function navierstokes_av_nflux(eq::EulerEquation)
+  @accepts        ldg_u, u, smoothed_mu
+  @accepts [bdry] bdry_ldg_u, bdry_u, bdry_smoothed_mu
+  @accepts [bdry] nflx_q, nflx_E, nx
+
+  G      = 4 * smoothed_mu * ldg_u + smoothed_mu * ldg_u
+  bdry_G = 4 * bdry_smoothed_mu * bdry_ldg_u + bdry_smoothed_mu * bdry_ldg_u
+  nflx       = nx*G
+  bdry_nflx  = nx*bdry_G
+  nflx_q    += 0.5 * (nflx + bdry_nflx)
+
+  nflx       = nx*G*u
+  bdry_nflx  = nx*bdry_G*bdry_u
+  nflx_E    += 0.5 * (nflx + bdry_nflx)
+
+  @returns [bdry] nflx_q, nflx_E
+end