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--- docs:classes:dns [2009/02/16 18:14]
gibson
+++ docs:classes:dns [2010/02/02 07:55] (current)
@@ Line 23: / Line 23: @@
   FlowField u(Nx,Ny,Nz,3,Lx,Lz,a,b);     // velocity
-  FlowField p(Nx,Ny,Nz,3,Lx,Lz,a,b);     // pressure
+  FlowField p(Nx,Ny,Nz,1,Lx,Lz,a,b);     // pressure
   DNS dns(u, nu, dt, flags);
   for (Real t=T0; t<=T1; t += N*dt) {
-     dns.advance(u,p,N);
+     dns.advance(u,p,N);                 // advance u,p forward N steps of length dt
      ...
   }
@@ Line 36: / Line 36: @@
 ===== DNSFlags =====
+The differents DNS flags are :
+  * baseflow
+  * timestepping
+  * initstepping
+  * nonlinearity
+  * dealiasing
+  * taucorrection
+  * constraint
+  * verbosity
 ===== Time-stepping algorithms =====
+The DNS class implements seven different time-stepping algorithms. (The default is SBDF3.)
+  * **CNFE1 or SBDF1**: 1st-order Crank-Nicolson, Foward-Euler or 1st-order Semi-implicit Back-
+ward Differentiation Formula –two names for the same algorithm. This algorithms is extremely
+simple and needs no initialization need, but its 1st-order error scaling makes it practically worth-
+less, except for initializing other algorithms.
+  * **CNAB2** 2nd-order Crank-Nicolson, Adams-Bashforth. A popular algorithm, but higher-frequency
+modes are poorly damped. Requires one initialization step. Zang warns against us-
+ing CNAB2 in combination with Rotational nonlinearity unless the high-frequency modes are
+dealiased. CNAB2 enforces zero-divergence at successive timesteps and momentum equa-
+tions halfway between successive time steps, which can lead to slowly decaying period-2dt os-
+cillation in the pressure field, unless pressure and velocity are initialized accurately.
+  * **CNRK2**: a three-substep, 2nd-order semi-implicit Crank-Nicolson, Runge-Kutta algorithm, devel-
+oped by Zang and Hussaini and but implemented in Channelflow from the Peyret’s exposition
+. According to Peyret, Zang and Hussaini observed 3rd-order scaling for this algorithm applied
+to low-viscosity flows, even though it is theoretically 2nd-order. Numerical tests in Channelflow
+show 2nd-order scaling for velocity fields at Re = 103 − 104 , and 1st-order scaling for pressure,
+due to a phase error in the pressure field. CNRK2 requires no initialization.
+  * **SMRK2**: a three-substep, 2nd-order semi-implicit Runge-Kutta developed by Spalart, Moser, and
+Rogers. Identical characteristics as CNRK2, including observed 2nd-order scaling consistent
+with theory, contrary to authors’ claim of 3rd-order scaling, and 1st-order phase error in pressure.
+Requires no initialization.
+  *** SBDF2, SBDF3, SBDF4**: 2nd, 3rd, and 4th-order Semi-implicit Backward Differentiation
+Formulae, requiring 1,2, and 3 initialization steps. I have found the SBDF schemes to be the
+best-behaved of the lot. When solving un+1 and pn+1 , SBDF schemes enforce divergence and
+momentum equations at tn+1 . This strongly implicit formulation poduces strong damping for
+high-frequency modes and results in pressure field as accurate as the velocity field. SBDF3 is par-
+ticularly good: it has the strongest asympotitc decay of all 3rd-order implicit-explicit linear multi-
+step schemes. For these reasons, SBDF3 is the default value of flags.timestepping. Peyret
+terms these algorithms AB/BDEk (kth-order Adams-Bashforth Backward-Differentiation).
+To summerize : **CNFE1, CNAB2, CNKR2, SMRK2, SBDF1, SBDF2, SBDF3, SBDF4**
 ===== Nonlinearity =====
-===== Base flow =====
+The nonlinear term in the Navier-Stokes calculation can be computed in a
+number of forms that are equivalent in continuous mathematics but slightly different when computed
+with spectral expansions and collocation. The default is SkewSymmetric.
-===== Mean constraint =====
+  * **Rotational:** Fast but generates high-frequency errors unless dealiased
+  * **SkewSymmetric:** Comparatively expensive to compute compared to Rotational
+  * **Convective**
+  * **Divergence**
+  * **Alternating** convection/divergence an alternating time steps. A cheap approximation to SkewSymmetric, which is an average of the convective and divergence forms. Not yet analyzed how the alternating nonlinearity method interacts with multistepping algorithms.
+  * **Linearized** about the base flow.
+===== Base flow =====
+  * **Zeros**
+  * **PlaneCouette** : plane Couette mean velocity profile <latex> y </latex>
+  * **Parabolic** : plane Poiseuille mean velocity profile <latex> 1-y^2 </latex>
+===== Mean constraint =====
+Periodic channel flows satisfy the Navier-Stokes equations with either the **bulk velocity** or the **spatial-mean pressure gradient** set as an external constraint. This flag sets which constraint is to be enforced. DNS’s default behavior determines the spatial-mean pressure gradient or bulk velocity from the fluctuation’s initial condition u and matches this as a fixed constraint at each time step. DNS can match time-varying constraints as well.

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