diff --git a/doc/ADI_scheme.org b/doc/ADI_scheme.org index e0d2abd..1124a92 100644 --- a/doc/ADI_scheme.org +++ b/doc/ADI_scheme.org @@ -1,7 +1,7 @@ -#+TITLE: Numerical solution of diffusion equation in 2D with ADI Scheme +#+TITLE: Finite Difference Schemes for the numerical solution of heterogeneous diffusion equation in 2D #+LaTeX_CLASS_OPTIONS: [a4paper,10pt] #+LATEX_HEADER: \usepackage{fullpage} -#+LATEX_HEADER: \usepackage{amsmath, systeme} +#+LATEX_HEADER: \usepackage{amsmath, systeme, cancel, xcolor} #+OPTIONS: toc:nil @@ -308,8 +308,8 @@ form: * Heterogeneous diffusion -If the diffusion coefficient $\alpha$ is spatially variable, [[eqn:1]] can -be rewritten: +If the diffusion coefficient $\alpha$ is spatially variable, equation +[[eqn:1]] can be rewritten as: #+NAME: eqn:hetdiff \begin{align} @@ -391,8 +391,8 @@ concentrations. ** Direct discretization As noted in literature (LeVeque and Numerical Recipes) a better way is -to discretize directly the physical problem ([[eqn:hetdiff]]) at points -halfway between grid points: +to discretize directly the physical problem (eq. [[eqn:hetdiff]]) at +points halfway between grid points: \begin{align*} \begin{cases} @@ -401,15 +401,18 @@ halfway between grid points: \end{cases} \end{align*} -\noindent A further differentiation gives us the centered +\noindent A further differentiation gives us the spatially centered approximation of $\frac{\partial}{\partial x} \left(\alpha(x) \frac{\partial C }{\partial x}\right)$: -\begin{align*} +#+NAME: eqn:CS_het +\begin{equation} +\begin{aligned} \frac{\partial}{\partial x} \left(\alpha(x) \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]\\ &\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] -\end{align*} +\end{aligned} +\end{equation} \noindent The ADI scheme with this approach becomes: @@ -439,3 +442,98 @@ explicit terms, the two sweeps become: \end{aligned} \right. \end{equation} + +\bigskip + +\noindent The "interblock" diffusion coefficients $\alpha_{i+1/2,j}$ +can be arithmetic mean: + +\[ +\displaystyle \alpha_{i+1/2, j} = \displaystyle \frac{\alpha_{i+1, j} + \alpha_{i, j}}{2} +\] + +\noindent or the harmonic mean: + +\[ +\displaystyle \alpha_{i+1/2, j} = \displaystyle \frac{2}{\frac{1}{\alpha_{i+1, j}} + \frac{1}{\alpha_{i, j}}} +\] + + + +\pagebreak + +* Explicit scheme for 2D heterogeneous diffusion + +A classical explicit FTCS scheme (forward in time, central in space) +for 2D heterogeneous diffusion can be expressed simply leveraging the +discretization of equation [[eqn:CS_het]]: + +#+NAME: eqn:2DHeterFTCS +\begin{equation} +\begin{aligned} +\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] + \\ + & \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] +\end{aligned} +\end{equation} +\noindent where in the RHS only the known concentrations at time $t$ +appear. Rearranging the terms, we get: + +#+NAME: eqn:2DHeterFTCS_final +\begin{equation} +\begin{aligned} +C_{i,j}^{t+1} = & C^t_{i,j} +\\ + & \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] + \\ + & \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] +\end{aligned} +\end{equation} + +The Courant-Friedrichs-Lewy stability criterion for this scheme reads: +#+NAME: eqn:CFL2DFTCS +\begin{equation} +\Delta t \leq \frac{(\Delta x^2, \Delta y^2)}{2 \max(\alpha_{i,j})} +\end{equation} + +** Boundary conditions + +In analogy to the treatment of the 1D homogeneous FTCS scheme (cfr +section 1), we need to differentiate the domain boundaries ($i=0$ and +$i=n_x$; the same applies to $j$ of course) accounting for the +discrepancy in the discretization. + +For the zero-th (left) cell, whose center is at $x=dx/2$, we can +evaluate the left gradient with the left boundary using such distance, +calling $l$ the numerical value of a constant boundary condition, +equation [[eqn:CS_het]] becomes: + +#+NAME: eqn:2D_FTCS_left +\begin{equation} +\begin{aligned} +\frac{\partial}{\partial x} \left(\alpha(x) +\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]\\ +&\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] +\end{aligned} +\end{equation} + +\noindent Similarly, for $i=n_x$, + +#+NAME: eqn:2D_FTCS_right +\begin{equation} +\begin{aligned} +\frac{\partial}{\partial x} \left(\alpha(x) +\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]\\ +&\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] +\end{aligned} +\end{equation} + +If on the right boundary we have *closed* or Neumann condition, the +left derivative becomes zero and we are left with: + +#+NAME: eqn:2D_FTCS_rightclosed +\begin{equation} +\begin{aligned} +\frac{\partial}{\partial x} \left(\alpha(x) +\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]\\ +&\displaystyle =\frac{\alpha_{i-1/2}}{\Delta x^2} (C_{i-1} - C_i) +\end{aligned} +\end{equation} + diff --git a/doc/ADI_scheme.pdf b/doc/ADI_scheme.pdf index 41b0f4d..7a2fddb 100644 Binary files a/doc/ADI_scheme.pdf and b/doc/ADI_scheme.pdf differ diff --git a/scripts/Adi2D_Reference.R b/scripts/Adi2D_Reference.R index 9b0bdd1..947ddc8 100644 --- a/scripts/Adi2D_Reference.R +++ b/scripts/Adi2D_Reference.R @@ -1,4 +1,4 @@ -## Time-stamp: "Last modified 2023-01-05 17:52:55 delucia" +## Time-stamp: "Last modified 2023-05-11 17:31:41 delucia" ## Brutal implementation of 2D ADI scheme ## Square NxN grid with dx=dy=1 @@ -272,8 +272,8 @@ ADIHetDir <- function(field, dt, iter, alpha) { for (it in seq(1, iter)) { for (i in seq(2, ny-1)) { - Aij <- cbind(harm(alpha[i,], alpha[i-1,]), harm(alpha[i,], alpha[i+1,])) - Bij <- cbind(harm(alpha[,i], alpha[,i-1]), harm(alpha[,i], alpha[,i+1])) + Aij <- cbind(colMeans(rbind(alpha[i,], alpha[i-1,])), colMeans(rbind(alpha[i,], alpha[i+1,]))) + Bij <- cbind(rowMeans(cbind(alpha[,i], alpha[,i-1])), rowMeans(cbind(alpha[,i], alpha[,i+1]))) tmpX[i,] <- SweepByRowHetDir(i, res, dt=dt, Aij, Bij) } resY <- t(tmpX) @@ -321,22 +321,27 @@ SweepByRowHetDir <- function(i, field, dt, Aij, Bij) { ## adi2 <- ADI(n=51, dt=10, iter=200, alpha=1E-3) ## ref2 <- DoRef(n=51, alpha=1E-3, dt=10, iter=200) -n <- 5 +n <- 51 field <- matrix(0, n, n) alphas <- matrix(1E-5*runif(n*n, 1,2), n, n) -alphas1 <- matrix(3E-5, n, 25) -alphas2 <- matrix(1E-5, n, 26) -alphas <- cbind(alphas1, alphas2) +## dim(field) +## dim(alphas) +## all.equal(dim(field), dim(alphas)) + +## alphas1 <- matrix(3E-5, n, 25) +## alphas2 <- matrix(1E-5, n, 26) + +## alphas <- cbind(alphas1, alphas2) ## for (i in seq(1,nrow(alphas))) ## alphas[i,] <- seq(1E-7,1E-3, length=n) #diag(alphas) <- rep(1E-2, n) -adih <- ADIHetDir(field=field, dt=10, iter=200, alpha=alphas) -adi2 <- ADI(n=n, dt=10, iter=200, alpha=1E-5) +adih <- ADIHetDir(field=field, dt=20, iter=500, alpha=alphas) +adi2 <- ADI(n=n, dt=20, iter=500, alpha=1E-5) par(mfrow=c(1,3)) @@ -347,6 +352,17 @@ plot(adih[[length(adih)]], adi2[[length(adi2)]], pch=4, log="xy") abline(0,1) +cchet <- lapply(adih, round, digits=6) +cchom <- lapply(adi2, round, digits=6) + +plot(cchet[[length(cchet)]], cchom[[length(cchom)]], pch=4, log="xy", xlim=c(1e-6,1), ylim=c(1e-6,1)) +abline(0,1) + +cchet[[500]] + +str(adih) + + sapply(adih, sum) sapply(adi2, sum) @@ -360,3 +376,108 @@ image(adih[[length(adih)]]) points(0.5,0.5, col="red",pch=4) options(width=110) + + +FTCS_2D <- function(field, dt, iter, alpha) { + + if (!all.equal(dim(field), dim(alpha))) + stop("field and alpha are not matrix") + + ## now both field and alpha must be nx*ny matrices + nx <- ncol(field) + ny <- nrow(field) + dx <- dy <- 1 + + ## find out the center of the grid to apply conc=1 + cenx <- ceiling(nx/2) + ceny <- ceiling(ny/2) + field[cenx, ceny] <- 1 + + ## prepare containers for computations and outputs + tmp <- res <- field + + cflt <- 1/max(alpha)/4 + cat(":: CFL allowable time step: ", cflt,"\n") + + ## inner iterations + inner <- floor(dt/cflt) + if (inner == 0) { + ## dt < cflt, no inner iterations + inner <- 1 + tsteps <- dt + cat(":: No inner iter. required\n") + } else { + tsteps <- c(rep(cflt, inner), dt-inner*cflt) + cat(":: Number of inner iter. required: ", inner,"\n") + } + + + out <- vector(mode="list", length=iter) + + for (it in seq(1, iter)) { + cat(":: outer it: ", it) + + for (innerit in seq_len(inner)) { + for (i in seq(2, ny-1)) { + for (j in seq(2, nx-1)) { + ## tmp[i,j] <- res[i,j] + + ## + tsteps[innerit]/dx/dx * (res[i+1,j]*mean(alpha[i+1,j],alpha[i,j]) - + ## res[i,j] *(mean(alpha[i+1,j],alpha[i,j])+mean(alpha[i-1,j],alpha[i,j])) + + ## res[i-1,j]*mean(alpha[i-1,j],alpha[i,j])) + + ## + tsteps[innerit]/dy/dy * (res[i,j+1]*mean(alpha[i,j+1],alpha[i,j]) - + ## res[i,j] *(mean(alpha[i,j+1],alpha[i,j])+mean(alpha[i,j-1],alpha[i,j])) + + ## res[i,j-1]*mean(alpha[i,j-1],alpha[i,j])) + tmp[i,j] <- res[i,j] + + + tsteps[innerit]/dx/dx * ((res[i+1,j]-res[i,j]) * mean(alpha[i+1,j],alpha[i,j]) - + (res[i,j]-res[i-1,j]) * mean(alpha[i-1,j],alpha[i,j])) + + + tsteps[innerit]/dx/dx * ((res[i,j+1]-res[i,j]) * mean(alpha[i,j+1],alpha[i,j]) - + (res[i,j]-res[i,j-1]) * mean(alpha[i,j-1],alpha[i,j])) + } + } + ## swap back tmp to res for the next inner iteration + res <- tmp + } + cat("- done\n") + ## at end of inner it we store + out[[it]] <- res + } + + return(out) +} + +## testing that FTCS with homog alphas reverts to ADI/Reference sim +n <- 51 +field <- matrix(0, n, n) +alphas <- matrix(1E-3, n, n) + +adi2 <- ADI(n=51, dt=100, iter=20, alpha=1E-3) + +ref <- DoRef(n=51, alpha=1E-3, dt=100, iter=20) + +adihet <- ADIHetDir(field=field, dt=100, iter=20, alpha=alphas) + +ftcsh <- FTCS_2D(field=field, dt=100, iter=20, alpha=alphas) + + +par(mfrow=c(2,4)) +image(ref, main="Reference ODE.2D") +points(0.5,0.5, col="red",pch=4) +image(ftcsh[[length(ftcsh)]], main="FTCS 2D") +points(0.5,0.5, col="red",pch=4) +image(adihet[[length(adihet)]], main="ADI Heter.") +points(0.5,0.5, col="red",pch=4) +image(adi2[[length(adi2)]], main="ADI Homog.", col=terrain.colors(12)) +points(0.5,0.5, col="red",pch=4) +plot(ftcsh[[length(ftcsh)]], ref, pch=4, log="xy", xlim=c(1E-16, 1), ylim=c(1E-16, 1), + main = "FTCS_2D vs ref", xlab="FTCS 2D", ylab="Reference") +abline(0,1) +plot(ftcsh[[length(ftcsh)]], adihet[[length(adihet)]], pch=4, log="xy", xlim=c(1E-16, 1), ylim=c(1E-16, 1), + main = "FTCS_2D vs ADI Het", xlab="FTCS 2D", ylab="ADI 2D Heter.") +abline(0,1) +plot(ftcsh[[length(ftcsh)]], adi2[[length(adi2)]], pch=4, log="xy", xlim=c(1E-16, 1), ylim=c(1E-16, 1), + main = "FTCS_2D vs ADI Hom", xlab="FTCS 2D", ylab="ADI 2D Hom.") +abline(0,1) +plot(adihet[[length(adihet)]], adi2[[length(adi2)]], pch=4, log="xy", xlim=c(1E-16, 1), ylim=c(1E-16, 1), + main = "ADI Het vs ADI Hom", xlab="ADI Het", ylab="ADI 2D Hom.") +abline(0,1) +