Numerical methodΒΆ
I integrate the following three equations in time:
\[\der{\wav{\ux}{\ix \iy}}{t}
=
\wav{h_x}{\ix \iy}
+
\frac{\mkx}{\mkx^2 + \mky^2}
\left(
\mkx \wav{h_x}{\ix \iy}
+
\mky \wav{h_y}{\ix \iy}
\right)
-
\frac{1}{Re} \left( \mkx^2 + \mky^2 \right) \wav{\ux}{\ix \iy},\]
\[\der{\wav{\uy}{\ix \iy}}{t}
=
\wav{h_y}{\ix \iy}
+
\frac{\mky}{\mkx^2 + \mky^2}
\left(
\mkx \wav{h_x}{\ix \iy}
+
\mky \wav{h_y}{\ix \iy}
\right)
-
\frac{1}{Re} \left( \mkx^2 + \mky^2 \right) \wav{\uy}{\ix \iy},\]
\[\der{\wav{T}{\ix \iy}}{t}
=
\wav{g}{\ix \iy}
-
\frac{1}{Re Sc} \left( \mkx^2 + \mky^2 \right) \wav{T}{\ix \iy},\]
where the non-linear terms are given by
\[\wav{h_x}{\ix \iy}
\equiv
-
I \mkx
\sum_{\ix_0^{\prime} + \ix_1^{\prime} = \ix}
\sum_{\iy_0^{\prime} + \iy_1^{\prime} = \iy}
\wav{\ux}{\ix_0^{\prime} \iy_0^{\prime}}
\wav{\ux}{\ix_1^{\prime} \iy_1^{\prime}}
-
I \mky
\sum_{\ix_0^{\prime} + \ix_1^{\prime} = \ix}
\sum_{\iy_0^{\prime} + \iy_1^{\prime} = \iy}
\wav{\uy}{\ix_0^{\prime} \iy_0^{\prime}}
\wav{\ux}{\ix_1^{\prime} \iy_1^{\prime}}
+
\wav{a_x}{\ix \iy},\]
\[\wav{h_y}{\ix \iy}
\equiv
-
I \mkx
\sum_{\ix_0^{\prime} + \ix_1^{\prime} = \ix}
\sum_{\iy_0^{\prime} + \iy_1^{\prime} = \iy}
\wav{\ux}{\ix_0^{\prime} \iy_0^{\prime}}
\wav{\uy}{\ix_1^{\prime} \iy_1^{\prime}}
-
I \mky
\sum_{\ix_0^{\prime} + \ix_1^{\prime} = \ix}
\sum_{\iy_0^{\prime} + \iy_1^{\prime} = \iy}
\wav{\uy}{\ix_0^{\prime} \iy_0^{\prime}}
\wav{\uy}{\ix_1^{\prime} \iy_1^{\prime}}
+
\wav{a_y}{\ix \iy},\]
\[\wav{g}{\ix \iy}
\equiv
-
I \mkx
\sum_{\ix_0^{\prime} + \ix_1^{\prime} = \ix}
\sum_{\iy_0^{\prime} + \iy_1^{\prime} = \iy}
\wav{\ux}{\ix_0^{\prime} \iy_0^{\prime}}
\wav{ T}{\ix_1^{\prime} \iy_1^{\prime}}
-
I \mky
\sum_{\ix_0^{\prime} + \ix_1^{\prime} = \ix}
\sum_{\iy_0^{\prime} + \iy_1^{\prime} = \iy}
\wav{\uy}{\ix_0^{\prime} \iy_0^{\prime}}
\wav{ T}{\ix_1^{\prime} \iy_1^{\prime}}.\]
Note
For now the external forcing terms \(a_i\) are omitted.