PDE Modeling

PDE Modeling, MA 461 Assignment 12: Invasion Fronts Due date: 04/19/2021 In this assignment we use COMSOL Multiphysics to investigate the 2-D axisymmetric version of invasion fronts. Email the instructor the mph file and upload your Report (see the end of the assignment) to Canvas. The Background In Section 3.4.4 we derive the 1D dimensionless Fisher equation ∂ 2n ∂n = + n(1 − n), ∂t ∂x2 (3.30) where n is the normalized population density in the location x at time t. Similarly, the 2D dimensionless Fisher equation is ∂ 2n ∂ 2n ∂n = + + n(1 − n), ∂t ∂x2 ∂y 2 (*) where the normalized population density n = n(x, y, t) depends on the location (x, y) and time t. In this assignment we study axisymmetric solutions of (*). As in Section 3.4.4, we use polar coordinates (r, θ) to replace (x, y) so n = n(r, θ, t). Next, we consider solutions depending on t and r only, n = n(r, t). The dimensionless equation is ∂ 2 n 1 ∂n ∂n = 2 + + n(1 − n). ∂t ∂r r ∂r (3.34) The Problem In the lecture notes we learn that unlike the 1D Fisher equation (3.30), the 2D axisymmetric equation (3.34) has an extra convection term. Because of this term (3.34) does not possess traveling wave solutions. If n(r, 0) = n0 (r) looks like the one in Figure 3.10 of the lecture notes, there will be logistic growth and diffusion, which lead to the formation of some sort of ‘wave front’. The front propagates with a ‘wave speed’ depending on r and is smaller than 2. As the front moves outward, eventually r becomes large and the convection term is no longer important. Therefore, the ‘speed’ approaches to 2 as t becomes large. In this assignment we use COMSOL Multiphysics to verify such a fact. In the lecture notes we also learn that for 1D Fisher equation (3.30), the traveling wave speed c in the time-asymptotic limit depends sensitively on the decay of n0 (x) to 0 as x → ∞. Suppose n0 (x) is not identically zero for large x but approaches zero like Ae−βx , where A and β are positive constants. Then the solution of (3.30), (3.31) approaches to a traveling wave with speed c = β +1/β if 0 < β < 1 and c = 2 if β ≥ 1. In the second part of this assignment we investigate the corresponding situation for the 2D axisymmetric equation (3.34). COMSOL Multiphysics Start Model Wizard and choose 1D Axisymmetric. Then select Mathematics −→ PDE Interfaces −→ Coefficient Form PDE. Click Add. In the Review Physics window set Dependent variables as n. Click Study. In the Select Study window choose Time Dependent and click Done. Click Untitled.mph (root). In the Settings window set Unit System as None. Use Definitions −→ Functions −→ Piecewise to define the initial function: ( 0.5 cos(r) if 0 ≤ r ≤ π2 , n0 (r) = (1) π < r ≤ 50. 0 2 Note that this is the 2D axisymmetric version of the initial data used to generate Figure 3.10 in the lecture notes. In the Piecewise window set Argument as r. Then define n0 accordingly. Plot the function to verify it. Add an Interval node to Geometry 1 and build the interval [0, 50] for r. In the Coefficient Form PDE Settings window expand Equation and set coefficients. You need to set f as n*(1-n) and β as −1/r (expanding Convection Coefficient). Set initial condition as n0(r) if n0 is the name of the piecewise function defined in (1). Choose the option Extremely fine and build the mesh. In the Settings window for Step 1: Time Dependent set Times as range(0,5,20). Then compute. The Report 1. Include in your report the plot from 1D Plot Group 1, together with your discussion on ‘wave front’ and ‘wave speed’. Also view 2D Plot Group 2 for each time step or view an animation of it. 2. In this part we explore 2D axisymmetric solutions of Fisher equation when the initial function n0 (r) is not identically zero for large r but approaches zero like Ae−βr , where A and β are positive constants. We consider the following two cases: (a) ( 0.5 cos(r) if 0 ≤ r ≤ π2 , n1 (r) = π 0.1e−1.1r < r ≤ 50 2 (2) ( 0.5 cos(r) if 0 ≤ r ≤ π2 , n2 (r) = π 0.1e−0.5r < r ≤ 50 2 (3) (b) The new initial functions n1 and n2 decay to 0 as r → ∞ with rates β = 1.1 and β = 0.5, respectively. (Note that these functions are not monotonically decreasing, and have a jump discontinuity at r = π/2.) Repeat the computation in Problem 1 with n0 replaced by n1 , then by n2 . Make a discussion on how the ‘wave speed’ depends on the value of β. Give an explanation why the speed depends on β in such way. Include your plots in the report.
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