As far as literature searches go in the direction of cataloguing finite difference methods explicitly, there appears to be no descriptive representation of reaction\u2013diffusion equations in their full three spatial dimensions. It is therefore our task (albeit tedious) to assemble the matrices and vectors for the model.<\/p>\n
We extend the two-dimensional operator formulation to the
\nthree-dimensional case by introducing an additional spatial index
\n\\( n \\in \\{0,1,\\dots,L\\} \\) with step size \\( \\Delta z \\).
\nThe flux operators \\( (\\epsilon^{+}_x + \\epsilon^{+}_y) \\) are augmented by
\nthe \\( z \\)-direction term \\( \\epsilon^{+}_z \\), so that the discrete diffusion operator
\nacting on the substrate \\( S \\) is given by the 3D seven-point stencil.<\/p>\n
\\[
\n\\begin{aligned}
\n(\\epsilon^{+}_x + \\epsilon^{+}_y + \\epsilon^{+}_z) S^k_{l,m,n}
\n&= a^{-}_{l,m,n;x}\\, S^{k+1}_{l-1,m,n}
\n + a^{+}_{l,m,n;x}\\, S^{k+1}_{l+1,m,n} \\\\
\n&\\quad + a^{-}_{l,m,n;y}\\, S^{k+1}_{l,m-1,n}
\n + a^{+}_{l,m,n;y}\\, S^{k+1}_{l,m+1,n} \\\\
\n&\\quad + a^{-}_{l,m,n;z}\\, S^{k+1}_{l,m,n-1}
\n + a^{+}_{l,m,n;z}\\, S^{k+1}_{l,m,n+1} \\\\
\n&\\quad + b_{l,m,n}\\, S^{k+1}_{l,m,n}.
\n\\end{aligned}
\n\\tag{1}
\n\\]<\/p>\n
\\[
\n\\begin{aligned}
\na^{-}_{l,m,n;x} &= \\frac{\\Delta t}{(\\Delta x)^2}\\,
\nD\\!\\left(\\frac{M^k_{l-1,m,n} + M^k_{l,m,n}}{2}\\right),\\\\
\na^{+}_{l,m,n;x} &= \\frac{\\Delta t}{(\\Delta x)^2}\\,
\nD\\!\\left(\\frac{M^k_{l,m,n} + M^k_{l+1,m,n}}{2}\\right),\\\\[6pt]
\na^{-}_{l,m,n;y} &= \\frac{\\Delta t}{(\\Delta y)^2}\\,
\nD\\!\\left(\\frac{M^k_{l,m-1,n} + M^k_{l,m,n}}{2}\\right),\\\\
\na^{+}_{l,m,n;y} &= \\frac{\\Delta t}{(\\Delta y)^2}\\,
\nD\\!\\left(\\frac{M^k_{l,m,n} + M^k_{l,m+1,n}}{2}\\right),\\\\[6pt]
\na^{-}_{l,m,n;z} &= \\frac{\\Delta t}{(\\Delta z)^2}\\,
\nD\\!\\left(\\frac{M^k_{l,m,n-1} + M^k_{l,m,n}}{2}\\right),\\\\
\na^{+}_{l,m,n;z} &= \\frac{\\Delta t}{(\\Delta z)^2}\\,
\nD\\!\\left(\\frac{M^k_{l,m,n} + M^k_{l,m,n+1}}{2}\\right).
\n\\end{aligned}
\n\\]<\/p>\n
\\[
\nb_{l,m,n} =
\n1 \u2013 \\left(
\na^{-}_{l,m,n;x} + a^{+}_{l,m,n;x}
\n+ a^{-}_{l,m,n;y} + a^{+}_{l,m,n;y}
\n+ a^{-}_{l,m,n;z} + a^{+}_{l,m,n;z}
\n\\right).
\n\\tag{2}
\n\\]<\/p>\n
\\[
\n\\mathcal{A}^k_Y =
\n\\begin{bmatrix}
\nA^k_Y & H^{+}_1 & 0 & \\cdots & 0 \\\\
\nH^{-}_2 & A^k_Y & H^{+}_2 & \\cdots & 0 \\\\
\n0 & H^{-}_3 & A^k_Y & \\ddots & \\vdots \\\\
\n\\vdots & \\ddots & \\ddots & \\ddots & H^{+}_{L-1} \\\\
\n0 & \\cdots & 0 & H^{-}_L & A^k_Y
\n\\end{bmatrix}.
\n\\tag{3}
\n\\]<\/p>\n
\\[
\n\\mathcal{M}^k =
\n\\begin{bmatrix}
\n\\mathcal{A}^k_S & \\mathcal{Z}^k_1 & 0 & \\mathcal{H}^k \\\\
\n\\mathcal{G}^k_1 & \\mathcal{A}^k_X & 0 & 0 \\\\
\n0 & \\mathcal{Z}^k_2 & \\mathcal{A}^k_I & 0 \\\\
\n\\mathcal{G}^k_2 & 0 & 0 & \\mathcal{A}^k_E
\n\\end{bmatrix}.
\n\\tag{4}
\n\\]<\/p>\n
\\[
\n\\mathcal{A}^k_Y =
\nI_z \\otimes A^k_Y + T_z \\otimes I_{xy},
\n\\tag{5}
\n\\]
\nwhere \\( I_z \\) is the identity in the \\(z\\)-direction,
\n\\( I_{xy} \\) is the identity in the \\((x,y)\\)-plane,
\nand \\( T_z \\) contains the coefficients \\( a^{\\pm}_{l,m,n;z} \\).
\nThis form preserves sparsity and allows direct use of sparse BLAS.<\/p>\n","protected":false},"excerpt":{"rendered":"
As far as literature searches go in the direction of cataloguing finite difference methods explicitly, there appears to be no descriptive representation<\/p>\n