1-D Advection, Diffusion and Reaction Modeling Project
Part 1. Forward Time Centered Space (2nd Order) Finite Difference
Due 24 April 2025
1. Diffusion Equation Solution. Write a FTCS model, with a domain length of 4m and a
total run time of at least 400 s. Take the diffusion coefficient to be 10-3 m2/s. Choose an appropriate space (e.g. 1 cm) and time discretization.
a. Simulate the evolution of an initial Gaussian plume centered in the domain, with peak concentration 10 [mg/l], and a length scale (σ ) of 0.2m. Use Dirichlet boundary conditions (C=0) and solve for the plume dynamics through space and time.
i. Plot C(X) at t=0, 100s, 200s, 300s, and 400s.
ii. Compare the plots above to an appropriate analytical model and discuss how well the numerical modeling is doing.
iii. Compute the diffusive flux and plot it vs X for t=0, 100s, 200s, 300s, and 400s. Comment.
iv. Compute the total mass in the domain and graph as a function of time.
Discuss. Try to run for a longer period of time, graph again and discuss.
b. Do as in 1a, but for IC of C(X,t=0)=0; BC of C(X=Left,t)= a specified flux of 0.01 mg/cm2/s, and C(X=Right,t)= 10mg/l.
i-iv. Construct analogous graphs as above and discuss
2. Advection-Diffusion Solution. Write a FTCS model, with a domain length of 6m and a total run time of at least 400 s. Take the diffusion coefficient to be 10-3 m2/s and the advective velocity to be 0.01 m/s. Choose an appropriate space (e.g. 1 cm) and time discretization.
a. Simulate the evolution of an initial Gaussian plume centered in the domain, with peak concentration 10 [mg/l], and a length scale of 0.2m. Use Dirichlet boundary conditions (C=0) and solve for the plume dynamics through space and time.
i. Plot C(X) at t=0, 100s, 200s, 300s, and 400s.
ii. Compare the plots above to an appropriate analytical model and discuss how well the numerical modeling is doing.
iii. Compute the total flux and plot it vs X for t= at t=0, 100s, 200s, 300s, and 400s. Comment on advective vs diffusive parts.
iv. Compute the total mass in the domain and graph as a function of time.
Discuss. Try to run for a longer period of time, graph and discuss.
b. Do as in 2a, but for IC of C(X, t=0)=0, and BC of C(X=Left, t)=10mg/l; C(X=Right, t)=0.
i-iv. Construct analogous graphs as above and discuss
c. Do as in 2b, but change left BC to a specified flux at the left boundary of 0.01 mg/l
i. Plot C(X) at t=0, 100s, 200s, 300s, and 400s.
ii. Plot C(left, t) for all time
3. Advection-Diffusion-Reaction Solution. Write a FTCS model, with a domain length of 6m and a total run time of at least 400 s. Take the diffusion coefficient to be 10-3 m2/s, the advective velocity to be 0.01 m/s, and the (sink) reaction constant to be k = - 0.003 [s-1]. Choose an appropriate space (e.g. 1 cm) and time discretization.
a. Simulate the evolution of an initial Gaussian plume centered in the domain, with peak concentration 10 [mg/l], and a length scale of 0.2m. Use Dirichlet boundary conditions (C=0) and solve for the plume dynamics through space and time.
i. Plot C(X) at t=0, 100s, 200s, 300s, and 400s.
ii. Compare the plots above to an appropriate analytical model and discuss how well the numerical modeling is doing.
iii. Compute the total flux and plot it vs X for t= at t=0, 100s, 200s, 300s, and 400s. Comment on advective vs diffusive parts.
iv. Compute the total mass in the domain and graph as a function of time.
Discuss. Try to run for a longer period of time, graph looks and discuss.
b. Do as in 3a, but for IC of C(X, t=0)=0, and BC of C(X=Left, t)=10mg/l; C(X=Right,
t)=0.
i-iv. Construct analogous graphs as above and discuss, including comparison to no reaction case.
c. Do as in 3b, but change left BC to a specified flux at the left boundary of 0.01 mg/l
i. Plot C(X) at t=0, 100s, 200s, 300s, and 400s.
ii. Compare to case with no reaction
Part 2
1-D Advection, Diffusion and Reaction Modeling Project
Part 2. Crank-Nicholson (2nd Order) Finite Difference
Due at end of semester
1. Perform. the same set of BC and IC runs as in part 1 of this project.
2. Experiment with the effect of time step on the solutions.
3. Compare this CN solution to the FTCS solution along side analytical results and comment on the findings.