From d986f862436d12862a55eabcb17451854d8d2e2d Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian.jauslin@rutgers.edu>
Date: Tue, 19 Sep 2023 11:51:55 -0400
Subject: [PATCH] Fix: no power 1/3 in enstrophy norm

---
 docs/nstrophy_doc.tex | 6 +++---
 src/navier-stokes.c   | 6 +++---
 2 files changed, 6 insertions(+), 6 deletions(-)

diff --git a/docs/nstrophy_doc.tex b/docs/nstrophy_doc.tex
index 3edbbe7..09efcba 100644
--- a/docs/nstrophy_doc.tex
+++ b/docs/nstrophy_doc.tex
@@ -177,16 +177,16 @@ It can be made by specifying the parameter {\tt adaptive\_norm}.
   \begin{equation}
     \|f\|:=\frac1{\mathcal N}\sqrt{\sum_k k^2|f_k|^2}
     ,\quad
-    \mathcal N:=\left(\frac{\sqrt{\sum_k k^2|\hat u_k^{(n)}|^2}+\sqrt{\sum_k k^2|\hat U_k^{(n)}|^2}}{\sum_k k^2|\hat u_k^{(n)}|^2}\right)^{\frac13}
+    \mathcal N:=\frac{\sqrt{\sum_k k^2|\hat u_k^{(n)}|^2}+\sqrt{\sum_k k^2|\hat U_k^{(n)}|^2}}{\sum_k k^2|\hat u_k^{(n)}|^2}
     .
   \end{equation}
   Doing so controls the error of the enstrophy through
   \begin{equation}
-    \mathcal N^2|\mathcal En(\hat u)-\mathcal En(\hat U)|\equiv|\|\hat u\|^2-\|\hat U\|^2|\leqslant \|\hat u-\hat U\|(\|\hat u\|+\|\hat U\|)
+    \frac1{\mathcal N^2}|\mathcal En(\hat u)-\mathcal En(\hat U)|\equiv|\|\hat u\|^2-\|\hat U\|^2|\leqslant \|\hat u-\hat U\|(\|\hat u\|+\|\hat U\|)
   \end{equation}
   so
   \begin{equation}
-    \mathcal N^2
+    \frac1{\mathcal N^2}
     |\mathcal En(\hat u)-\mathcal En(\hat U)|\leqslant
     \|\hat u-\hat U\|\frac1{\mathcal N}\left(\sqrt{\sum_k k^2|\hat u_k|^2}+\sqrt{\sum_k k^2|\hat U_k|^2}\right)
   \end{equation}
diff --git a/src/navier-stokes.c b/src/navier-stokes.c
index 9ab9126..dbdcd12 100644
--- a/src/navier-stokes.c
+++ b/src/navier-stokes.c
@@ -820,7 +820,7 @@ int ns_step_rkf45(
       }
     }
     err=sqrt(err);
-    relative=pow((sqrt(sumu)+sqrt(sumU))/sumu, 1./3);
+    relative=(sqrt(sumu)+sqrt(sumU))/sumu;
   }
   else{
     fprintf(stderr,"bug: unknown norm: %u, contact ian.jauslin@rutgers.edu\n", adaptive_norm);
@@ -966,7 +966,7 @@ int ns_step_rkbs32(
       }
     }
     err=sqrt(err);
-    relative=pow((sqrt(sumu)+sqrt(sumU))/sumu, 1./3);
+    relative=(sqrt(sumu)+sqrt(sumU))/sumu;
   }
   else{
     fprintf(stderr,"bug: unknown norm: %u, contact ian.jauslin@rutgers,edu\n", adaptive_norm);
@@ -1143,7 +1143,7 @@ int ns_step_rkdp54(
       }
     }
     err=sqrt(err);
-    relative=pow((sqrt(sumu)+sqrt(sumU))/sumu, 1./3);
+    relative=(sqrt(sumu)+sqrt(sumU))/sumu;
   }
   else{
     fprintf(stderr,"bug: unknown norm: %u, contact ian.jauslin@rutgers,edu\n", adaptive_norm);