Higher Order Flux Corrected Horizontal Transport.

components/omega/configs/Default.yml

@@ -32,6 +32,9 @@ Omega:
     FluxThicknessType: Center
     HorzTracerFluxOrder: 2
     VerticalTracerFluxLimiterEnable: true
+    HorzTracerFluxLimiterEnable: false
+    HorzTracerFluxLimiterBudgetsEnable: false
+    HorzTracerFluxLimiterMonotonicityCheckEnable: false
     VerticalTracerFluxOrder: 3
   SfcStress:
     InterpType: Isotropic

components/omega/doc/design/Tendencies.md

@@ -52,11 +52,22 @@ class Tendencies{
    VelocityHyperDiffOnEdge VelocityHyperDiff;
    SfcStressForcingOnEdge SfcStressForcing;
    BottomDragOnEdge ExplicitBottomDrag;
+   TracerHorzAdvOnCell TracerHorzAdv;
    TracerDiffOnCell TracerDiffusion;
    TracerHyperDiffOnCell TracerHyperDiff;
-   TracerHorzAdvOnCell TracerHorzAdv;
-   TracerHighOrderHorzAdvOnCell TracerHighOrderHorzAdv;
+   SurfaceTracerRestoringOnCell SurfaceTracerRestoring;
+
  private:
+   const HorzMesh *Mesh; ///< Pointer to horizontal mesh
+   VertCoord *VCoord;    ///< Pointer to vertical coordinate
+   VertAdv *VAdv;        ///< Pointer to vertical advection
+   CustomTendencyType CustomThicknessTend;
+   CustomTendencyType CustomVelocityTend;
+   Eos *EqState;          ///< Pointer to equation of state
+   PressureGrad *PGrad;   ///< Pointer to pressure gradient
+   VertMix *VMix;         ///< Pointer to vertical mixing
+   I4 NTracers;           ///< Number of tracers
+   TimeInterval TimeStep; ///< Time step
    static Tendencies *DefaultTendencies;
    static std::map<std::string, std::unique_ptr<Tendencies>> AllTendencies;
 };
@@ -106,7 +117,7 @@ void Tendencies::computeAllTendencies(const OceanState *State, const AuxilarySta
 ```
 The layer thickness tendencies will be computed with a method:
 ```c++
-void Tendencies::computeThicknessTendencies(const OceanState *State, const AuxilaryState *AuxState, int TimeLevel, int VelTimeLevel, TimeInstant Time);
+void Tendencies::computePseudoThicknessTendencies(const OceanState *State, const AuxilaryState *AuxState, int TimeLevel, int VelTimeLevel, TimeInstant Time);
 ```
 The normal velocity tendencies will be computed with a method:
 ```c++

components/omega/doc/devGuide/TendencyTerms.md

@@ -38,7 +38,6 @@ implemented:
 - `SfcStressForcingOnEdge`
 - `BottomDragOnEdge`
 - `TracerHorzAdvOnCell`
-- `TracerHighOrderHorzAdvOnCell`
 - `TracerDiffOnCell`
 - `TracerHyperDiffOnCell`
 - `SurfaceTracerRestoringOnCell`

components/omega/doc/userGuide/TendencyTerms.md

@@ -15,7 +15,6 @@ tendency terms are currently implemented:
 | VelocityDiffusionOnEdge | Laplacian horizontal mixing, defined on edges
 | VelocityHyperDiffOnEdge | biharmonic horizontal mixing, defined on edges
 | TracerHorzAdvOnCell | horizontal advection of thickness-weighted tracers
-| TracerHighOrderHorzAdvOnCell | second order horizontal advection of thickness-weighted tracers
 | TracerDiffOnCell | horizontal diffusion of thickness-weighted tracers
 | TracerHyperDiffOnCell | biharmonic horizontal mixing of thickness-weighted tracers
 | SfcStressForcingOnEdge | forcing by surface stress (e.g. wind), defined on edges
@@ -45,9 +44,11 @@ the currently available tendency terms:
 | | ViscDel4 | horizontal biharmonic mixing coefficient for normal velocity
 | | DivFactor | scale factor for the divergence term
 | TracerHorzAdvOnCell | TracerHorzAdvTendencyEnable | enable/disable term
-| | HorzTracerFluxOrder | 1 for standard linear advection
-| TracerHighOrderHorzAdvOnCell | TracerHorzAdvTendencyEnable | enable/disable term
-| | HorzTracerFluxOrder | 2 for second order advection algorithm
+| | HorzTracerFluxOrder | 2 for standard linear advection
+| | HorzTracerFluxOrder | 3 for second order advection algorithm
+| | HorzTracerFluxLimiterEnable | enable/disable monotonic flux corrected transport (FCT). Requires HorzTracerFluxOrder to be greater than 2.
+| | HorzTracerFluxLimiterBudgetsEnable | enable/disable budgets if FCT is enabled
+| | HorzTracerFluxLimiterMonotonicityCheckEnable | enable/disable check for mon-mononic values if FCT is enabled
 | TracerDiffOnCell | TracerDiffTendencyEnable | enable/disable term
 | | EddyDiff2 | horizontal diffusion coefficient
 | TracerHyperDiffOnCell | TracerHyperDiffTendencyEnable | enable/disable term
@@ -94,13 +95,13 @@ $$
 $$
 Where $u$ is $\mathbf{V}\cdot\mathbf{n}$ where $\mathbf{n}$ is the unit normal along the edge being evaluated and $F$ is the evaluation
 of $\psi$ for the elements on each side of the edge being evaluated.
-This is used when the TracerHorzAdvOnCell user option is active and HorzTracerFluxOrder is 1.
+This is used when the TracerHorzAdvOnCell user option is active and HorzTracerFluxOrder is 2.
 To make the comparison to higher order methods explicit, this first order method is equivalent to defining $\psi$ as a linear function,
 $\psi = c_0 + c_x x + c_y y$, between the two elements sharing the edge $i$ and taking the directed derivative along the edge.
 
 
 For higher order flux calculations, higher order derivatives of $\psi$ are needed and for the user option of
-TracerHorzAdvOnCell along with HorzTracerFluxOrder set to 2, a quadratic approximation of $\psi$ is used,
+TracerHorzAdvOnCell along with HorzTracerFluxOrder set to 3, a quadratic approximation of $\psi$ is used,
 $$
 \psi = c_0 + c_x x + c_y y + c_{xx} x^2 + c_{xy} xy + c_{yy} y^2,
 $$
@@ -140,6 +141,87 @@ is about order 1.7 as shown in  {numref}`tracer-higher-order-convergence`:
 Tracer higer order convergence example of a cosine bell advected on a sphere showing an order 1.71 convergence rate
 ```
 
+### Monotonic Flux Limiting Transport for Second Order Horizontal Advection
+
+It is well known that higher order (order 2 and above) scheme for numerically integrating fluxes suffer
+from dispersive "ripples" in the results particularly near steep gradients.  Lower order schemes
+produce no ripples but suffer from excessive numerical diffusion. Flux-corrected transport (FCT) is a
+technique which embodies the best of both schemes.
+
+The implementation of a monotonic flux-corrected transport (FCT) in Omega follows a standard algorithm as given in
+
+Zalesak, S. T. (1979). Fully multidimensional flux-corrected transport algorithms for fluids.
+Journal of Computational Physics, 31(3), 335–362. DOI: 10.1016/0021-9991(79)90051-2
+
+The procedure is as follows given the notation above:
+1. Compute $F^L_{i+1/2}(u\psi)$, the transportive flux by some low order scheme guarenteed to give
+monotonic (ripple-free) results.
+2. Compute  $F^H_{i+1/2}(u\psi)$, the transportative flux by some high order scheme.
+3. Define the "antififfusive flux":
+$$
+A_{i+1/2}(u\psi) = F^H_{i+1/2}(u\psi) - F^L_{i+1/2}(u\psi)
+$$
+4. Compute the updated low order ("transported and diffused") "td" solution:
+$$
+ w^{td}_i = w^n_i - \frac{1}{\Delta x}[F_{i+1/2}(u\psi) - F_{i-1/2}(u\psi)]
+$$
+5. Limit he $A_{i+1/2}(u\psi)$ in a manner such that $w^{n+1}$ as computed below is free of extrema
+not found in $w^{td}$ = $w^n$;
+$$
+A^C_{i+1/2}(u\psi) =  C_{i+1/2}(u\psi) A_{i+1/2}(u\psi), \quad  \quad   0 \leq C_{i+1/2} \leq 1
+$$
+6. Finally apply hte limited antidiffusive fluxes:
+$$
+ w^{n+1}_i = w^{td}_i - \frac{1}{\Delta x}[A^C_{i+1/2}(u\psi) - A^C_{i-1/2}(u\psi)]
+$$
+
+
+```{figure} images/higher_order_tracer_convergence_on_sphere_FCT.jpeg
+:name:  tracer-higher-order-convergence_FCT
+:align: center
+:width: 600 px
+Tracer higer order convergence example of a cosine bell advected on a sphere with FCT showing an order 2.23 convergence rate
+```
+
+
+When monotonic flux limiting is used then there are two diagnostic options that can be enabled,
+HorzTracerFluxLimiterBudgetsEnable and HorzTracerFluxLimiterMonotonicityCheckEnable. The
+HorzTracerFluxLimiterMonotonicityCheckEnable flag enables a post-transport check that the
+resulting values are indeed monotone as required by the algorithm and any discrepancies are
+printed along with information on where they occur.
+
+The HorzTracerFluxLimiterBudgetsEnable option enables the output of two diagnostic fields to the
+output mesh file.  One is the "FCTActiveTracerHorizontalAdvectionEdgeFlux" variable which is the
+edge flux across every side of every element. The other is the "FCTActiveTracerHorizontalAdvectionTendency"
+variable which is the cell-centered value of the advection tendency.
+
+
+### Monotonic Flux Limiting Transport Example of Limiting Ripple Effect
+
+The effect of monotonic higher order FCT horizontal transport verses just higher order horizontal transport
+can be seen in the Polaris test of the slotted cylinder. The cylinder is advected for one revolution under
+a constant velocity field. The result is advecting the solution back to the original configuration as
+an exact solution. The difference between the initial condition and the advected solution is the error
+in the advection scheme. This difference is shown in the following two plots for non-monotonic and monotonic
+advection.
+
+```{figure} images/higher_order_slotted_cylinder_non_monotonic_diff.jpg
+:name:  tracer-error-in-higher-order-convergence_non-monotonic
+:align: center
+:width: 600 px
+Error in a tracer higher order convergence example of a slotted cylinder advected on a sphere with the standard third order advection
+algorithm with no FCT flux correction. Notice the ringing due to the sharp edges of the tracer distribution.
+```
+
+```{figure} images/higher_order_slotted_cylinder_FCT_diff.jpg
+:name:  tracer-error-in-higher-order-convergence_FCT
+:align: center
+:width: 600 px
+Error Tracer higher order convergence example of a slotted cylinder advected on a sphere with FCT monotone
+flux correction. Notice the reduction in ringing around the sharp edges of the tracer distribution.
+```
+
+
 ## See Also
 
 Additional information on forcing (currently wind forcing and surface tracer