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MDL: Written down 2D explicit FTCS scheme in docs/ADI_scheme.org; rough implementation in scripts/Adi2D_Reference.R
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Marco De Lucia
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adb2e325be
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@ -1,7 +1,7 @@
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#+TITLE: Numerical solution of diffusion equation in 2D with ADI Scheme
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#+TITLE: Finite Difference Schemes for the numerical solution of heterogeneous diffusion equation in 2D
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#+LaTeX_CLASS_OPTIONS: [a4paper,10pt]
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#+LATEX_HEADER: \usepackage{fullpage}
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#+LATEX_HEADER: \usepackage{amsmath, systeme}
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#+LATEX_HEADER: \usepackage{amsmath, systeme, cancel, xcolor}
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#+OPTIONS: toc:nil
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@ -308,8 +308,8 @@ form:
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* Heterogeneous diffusion
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If the diffusion coefficient $\alpha$ is spatially variable, [[eqn:1]] can
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be rewritten:
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If the diffusion coefficient $\alpha$ is spatially variable, equation
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[[eqn:1]] can be rewritten as:
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#+NAME: eqn:hetdiff
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\begin{align}
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@ -391,8 +391,8 @@ concentrations.
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** Direct discretization
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As noted in literature (LeVeque and Numerical Recipes) a better way is
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to discretize directly the physical problem ([[eqn:hetdiff]]) at points
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halfway between grid points:
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to discretize directly the physical problem (eq. [[eqn:hetdiff]]) at
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points halfway between grid points:
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\begin{align*}
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\begin{cases}
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@ -401,15 +401,18 @@ halfway between grid points:
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\end{cases}
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\end{align*}
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\noindent A further differentiation gives us the centered
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\noindent A further differentiation gives us the spatially centered
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approximation of $\frac{\partial}{\partial x} \left(\alpha(x)
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\frac{\partial C }{\partial x}\right)$:
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\begin{align*}
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#+NAME: eqn:CS_het
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\begin{equation}
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\begin{aligned}
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\frac{\partial}{\partial x} \left(\alpha(x)
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\frac{\partial C }{\partial x}\right)(x_i) & \simeq \frac{1}{\Delta x}\left[\alpha_{i+1/2} \left( \frac{C_{i+1} -C_{i}}{\Delta x} \right) - \alpha_{i-1/2} \left( \frac{C_{i} -C_{i-1}}{\Delta x} \right) \right]\\
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&\displaystyle =\frac{1}{\Delta x^2} \left[ \alpha_{i+1/2}C_{i+1} - (\alpha_{i+1/2}+\alpha_{i-1/2}) C_{i} + \alpha_{i-1/2}C_{i-1}\right]
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\end{align*}
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\end{aligned}
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\end{equation}
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\noindent The ADI scheme with this approach becomes:
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@ -439,3 +442,98 @@ explicit terms, the two sweeps become:
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\end{aligned}
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\right.
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\end{equation}
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\bigskip
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\noindent The "interblock" diffusion coefficients $\alpha_{i+1/2,j}$
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can be arithmetic mean:
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\[
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\displaystyle \alpha_{i+1/2, j} = \displaystyle \frac{\alpha_{i+1, j} + \alpha_{i, j}}{2}
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\]
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\noindent or the harmonic mean:
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\[
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\displaystyle \alpha_{i+1/2, j} = \displaystyle \frac{2}{\frac{1}{\alpha_{i+1, j}} + \frac{1}{\alpha_{i, j}}}
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\]
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\pagebreak
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* Explicit scheme for 2D heterogeneous diffusion
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A classical explicit FTCS scheme (forward in time, central in space)
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for 2D heterogeneous diffusion can be expressed simply leveraging the
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discretization of equation [[eqn:CS_het]]:
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#+NAME: eqn:2DHeterFTCS
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\begin{equation}
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\begin{aligned}
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\frac{C_{i,j}^{t+1} - C_{i,j}^{t}}{\Delta t} = & \frac{1}{\Delta x^2} \left[ \alpha^x_{i+1/2, j}C^t_{i+1, j} - (\alpha^x_{i+1/2, j} + \alpha^x_{i-1/2, j}) C^t_{i,j} + \alpha^x_{i-1/2,j}C^t_{i-1,j}\right] + \\
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& \frac{1}{\Delta y^2} \left[ \alpha^y_{i, j+1/2}C^t_{i, j+1} - (\alpha^y_{i, j+1/2} + \alpha^y_{i, j-1/2}) C^t_{i,j} + \alpha^y_{i,j-1/2}C^t_{i,j-1}\right]
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\end{aligned}
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\end{equation}
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\noindent where in the RHS only the known concentrations at time $t$
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appear. Rearranging the terms, we get:
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#+NAME: eqn:2DHeterFTCS_final
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\begin{equation}
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\begin{aligned}
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C_{i,j}^{t+1} = & C^t_{i,j} +\\
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& \frac{\Delta t}{\Delta x^2} \left[ \alpha^x_{i+1/2, j}C^t_{i+1, j} - (\alpha^x_{i+1/2, j} + \alpha^x_{i-1/2, j}) C^t_{i,j} + \alpha^x_{i-1/2,j}C^t_{i-1,j}\right] + \\
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& \frac{\Delta t}{\Delta y^2} \left[ \alpha^y_{i, j+1/2}C^t_{i, j+1} - (\alpha^y_{i, j+1/2} + \alpha^y_{i, j-1/2}) C^t_{i,j} + \alpha^y_{i,j-1/2}C^t_{i,j-1}\right]
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\end{aligned}
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\end{equation}
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The Courant-Friedrichs-Lewy stability criterion for this scheme reads:
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#+NAME: eqn:CFL2DFTCS
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\begin{equation}
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\Delta t \leq \frac{(\Delta x^2, \Delta y^2)}{2 \max(\alpha_{i,j})}
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\end{equation}
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** Boundary conditions
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In analogy to the treatment of the 1D homogeneous FTCS scheme (cfr
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section 1), we need to differentiate the domain boundaries ($i=0$ and
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$i=n_x$; the same applies to $j$ of course) accounting for the
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discrepancy in the discretization.
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For the zero-th (left) cell, whose center is at $x=dx/2$, we can
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evaluate the left gradient with the left boundary using such distance,
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calling $l$ the numerical value of a constant boundary condition,
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equation [[eqn:CS_het]] becomes:
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#+NAME: eqn:2D_FTCS_left
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\begin{equation}
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\begin{aligned}
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\frac{\partial}{\partial x} \left(\alpha(x)
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\frac{\partial C }{\partial x}\right)(x_0) & \simeq \frac{1}{\Delta x}\left[\alpha_{i+1/2} \left( \frac{C_{i+1} -C_{i}}{\Delta x} \right) - \alpha_{i} \left( \frac{C_{i} - l }{\frac{\Delta x}{2}} \right) \right]\\
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&\displaystyle =\frac{1}{\Delta x^2} \left[ \alpha_{i+1/2}C_{i+1} - (\alpha_{i+1/2}+ 2\alpha_i) C_{i} + 2 \alpha_{i}\cdot l\right]
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\end{aligned}
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\end{equation}
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\noindent Similarly, for $i=n_x$,
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#+NAME: eqn:2D_FTCS_right
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\begin{equation}
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\begin{aligned}
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\frac{\partial}{\partial x} \left(\alpha(x)
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\frac{\partial C }{\partial x}\right)(x_n) & \simeq \frac{1}{\Delta x}\left[\alpha_{i} \left( \frac{r -C_{i}}{\frac{\Delta x}{2}} \right) - \alpha_{i-1/2} \left( \frac{C_{i} - C_{i-1}} {\Delta x} \right) \right]\\
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&\displaystyle =\frac{1}{\Delta x^2} \left[ 2 \alpha_{i} r - (\alpha_{i+1/2}+ 2\alpha_i) C_{i} + \alpha_{i-1/2}\cdot C_{i-1}\right]
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\end{aligned}
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\end{equation}
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If on the right boundary we have *closed* or Neumann condition, the
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left derivative becomes zero and we are left with:
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#+NAME: eqn:2D_FTCS_rightclosed
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\begin{equation}
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\begin{aligned}
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\frac{\partial}{\partial x} \left(\alpha(x)
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\frac{\partial C }{\partial x}\right)(x_n) & \simeq \frac{1}{\Delta x}\left[\cancel{\alpha_{i+1/2} \left( \frac{C_{i+1} -C_{i}}{\Delta x} \right)} - \alpha_{i-1/2} \left( \frac{C_{i} -C_{i-1}}{\Delta x} \right) \right]\\
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&\displaystyle =\frac{\alpha_{i-1/2}}{\Delta x^2} (C_{i-1} - C_i)
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\end{aligned}
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\end{equation}
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Binary file not shown.
@ -1,4 +1,4 @@
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## Time-stamp: "Last modified 2023-01-05 17:52:55 delucia"
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## Time-stamp: "Last modified 2023-05-11 17:31:41 delucia"
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## Brutal implementation of 2D ADI scheme
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## Square NxN grid with dx=dy=1
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@ -272,8 +272,8 @@ ADIHetDir <- function(field, dt, iter, alpha) {
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for (it in seq(1, iter)) {
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for (i in seq(2, ny-1)) {
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Aij <- cbind(harm(alpha[i,], alpha[i-1,]), harm(alpha[i,], alpha[i+1,]))
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Bij <- cbind(harm(alpha[,i], alpha[,i-1]), harm(alpha[,i], alpha[,i+1]))
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Aij <- cbind(colMeans(rbind(alpha[i,], alpha[i-1,])), colMeans(rbind(alpha[i,], alpha[i+1,])))
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Bij <- cbind(rowMeans(cbind(alpha[,i], alpha[,i-1])), rowMeans(cbind(alpha[,i], alpha[,i+1])))
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tmpX[i,] <- SweepByRowHetDir(i, res, dt=dt, Aij, Bij)
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}
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resY <- t(tmpX)
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@ -321,22 +321,27 @@ SweepByRowHetDir <- function(i, field, dt, Aij, Bij) {
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## adi2 <- ADI(n=51, dt=10, iter=200, alpha=1E-3)
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## ref2 <- DoRef(n=51, alpha=1E-3, dt=10, iter=200)
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n <- 5
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n <- 51
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field <- matrix(0, n, n)
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alphas <- matrix(1E-5*runif(n*n, 1,2), n, n)
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alphas1 <- matrix(3E-5, n, 25)
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alphas2 <- matrix(1E-5, n, 26)
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alphas <- cbind(alphas1, alphas2)
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## dim(field)
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## dim(alphas)
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## all.equal(dim(field), dim(alphas))
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## alphas1 <- matrix(3E-5, n, 25)
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## alphas2 <- matrix(1E-5, n, 26)
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## alphas <- cbind(alphas1, alphas2)
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## for (i in seq(1,nrow(alphas)))
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## alphas[i,] <- seq(1E-7,1E-3, length=n)
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#diag(alphas) <- rep(1E-2, n)
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adih <- ADIHetDir(field=field, dt=10, iter=200, alpha=alphas)
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adi2 <- ADI(n=n, dt=10, iter=200, alpha=1E-5)
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adih <- ADIHetDir(field=field, dt=20, iter=500, alpha=alphas)
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adi2 <- ADI(n=n, dt=20, iter=500, alpha=1E-5)
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par(mfrow=c(1,3))
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@ -347,6 +352,17 @@ plot(adih[[length(adih)]], adi2[[length(adi2)]], pch=4, log="xy")
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abline(0,1)
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cchet <- lapply(adih, round, digits=6)
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cchom <- lapply(adi2, round, digits=6)
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plot(cchet[[length(cchet)]], cchom[[length(cchom)]], pch=4, log="xy", xlim=c(1e-6,1), ylim=c(1e-6,1))
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abline(0,1)
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cchet[[500]]
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str(adih)
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sapply(adih, sum)
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sapply(adi2, sum)
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@ -360,3 +376,108 @@ image(adih[[length(adih)]])
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points(0.5,0.5, col="red",pch=4)
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options(width=110)
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FTCS_2D <- function(field, dt, iter, alpha) {
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if (!all.equal(dim(field), dim(alpha)))
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stop("field and alpha are not matrix")
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## now both field and alpha must be nx*ny matrices
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nx <- ncol(field)
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ny <- nrow(field)
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dx <- dy <- 1
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## find out the center of the grid to apply conc=1
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cenx <- ceiling(nx/2)
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ceny <- ceiling(ny/2)
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field[cenx, ceny] <- 1
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## prepare containers for computations and outputs
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tmp <- res <- field
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cflt <- 1/max(alpha)/4
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cat(":: CFL allowable time step: ", cflt,"\n")
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## inner iterations
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inner <- floor(dt/cflt)
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if (inner == 0) {
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## dt < cflt, no inner iterations
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inner <- 1
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tsteps <- dt
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cat(":: No inner iter. required\n")
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} else {
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tsteps <- c(rep(cflt, inner), dt-inner*cflt)
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cat(":: Number of inner iter. required: ", inner,"\n")
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}
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out <- vector(mode="list", length=iter)
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for (it in seq(1, iter)) {
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cat(":: outer it: ", it)
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for (innerit in seq_len(inner)) {
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for (i in seq(2, ny-1)) {
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for (j in seq(2, nx-1)) {
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## tmp[i,j] <- res[i,j] +
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## + tsteps[innerit]/dx/dx * (res[i+1,j]*mean(alpha[i+1,j],alpha[i,j]) -
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## res[i,j] *(mean(alpha[i+1,j],alpha[i,j])+mean(alpha[i-1,j],alpha[i,j])) +
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## res[i-1,j]*mean(alpha[i-1,j],alpha[i,j])) +
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## + tsteps[innerit]/dy/dy * (res[i,j+1]*mean(alpha[i,j+1],alpha[i,j]) -
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## res[i,j] *(mean(alpha[i,j+1],alpha[i,j])+mean(alpha[i,j-1],alpha[i,j])) +
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## res[i,j-1]*mean(alpha[i,j-1],alpha[i,j]))
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tmp[i,j] <- res[i,j] +
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+ tsteps[innerit]/dx/dx * ((res[i+1,j]-res[i,j]) * mean(alpha[i+1,j],alpha[i,j]) -
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(res[i,j]-res[i-1,j]) * mean(alpha[i-1,j],alpha[i,j])) +
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+ tsteps[innerit]/dx/dx * ((res[i,j+1]-res[i,j]) * mean(alpha[i,j+1],alpha[i,j]) -
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(res[i,j]-res[i,j-1]) * mean(alpha[i,j-1],alpha[i,j]))
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}
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}
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## swap back tmp to res for the next inner iteration
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res <- tmp
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}
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cat("- done\n")
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## at end of inner it we store
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out[[it]] <- res
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}
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return(out)
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}
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## testing that FTCS with homog alphas reverts to ADI/Reference sim
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n <- 51
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field <- matrix(0, n, n)
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alphas <- matrix(1E-3, n, n)
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adi2 <- ADI(n=51, dt=100, iter=20, alpha=1E-3)
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ref <- DoRef(n=51, alpha=1E-3, dt=100, iter=20)
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adihet <- ADIHetDir(field=field, dt=100, iter=20, alpha=alphas)
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ftcsh <- FTCS_2D(field=field, dt=100, iter=20, alpha=alphas)
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par(mfrow=c(2,4))
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image(ref, main="Reference ODE.2D")
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points(0.5,0.5, col="red",pch=4)
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image(ftcsh[[length(ftcsh)]], main="FTCS 2D")
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points(0.5,0.5, col="red",pch=4)
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image(adihet[[length(adihet)]], main="ADI Heter.")
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points(0.5,0.5, col="red",pch=4)
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image(adi2[[length(adi2)]], main="ADI Homog.", col=terrain.colors(12))
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points(0.5,0.5, col="red",pch=4)
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plot(ftcsh[[length(ftcsh)]], ref, pch=4, log="xy", xlim=c(1E-16, 1), ylim=c(1E-16, 1),
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main = "FTCS_2D vs ref", xlab="FTCS 2D", ylab="Reference")
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abline(0,1)
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plot(ftcsh[[length(ftcsh)]], adihet[[length(adihet)]], pch=4, log="xy", xlim=c(1E-16, 1), ylim=c(1E-16, 1),
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main = "FTCS_2D vs ADI Het", xlab="FTCS 2D", ylab="ADI 2D Heter.")
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abline(0,1)
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plot(ftcsh[[length(ftcsh)]], adi2[[length(adi2)]], pch=4, log="xy", xlim=c(1E-16, 1), ylim=c(1E-16, 1),
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main = "FTCS_2D vs ADI Hom", xlab="FTCS 2D", ylab="ADI 2D Hom.")
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abline(0,1)
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plot(adihet[[length(adihet)]], adi2[[length(adi2)]], pch=4, log="xy", xlim=c(1E-16, 1), ylim=c(1E-16, 1),
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main = "ADI Het vs ADI Hom", xlab="ADI Het", ylab="ADI 2D Hom.")
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abline(0,1)
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