Amme5202 Assignment 1: Advection And Diffusion Due 5pm Frida

Amme5202 Assignment 1 Advection And Diffusiondue 5pm Friday Week 7

Develop three finite volume solvers to analyze the two-dimensional advection-diffusion equation for a scalar quantity in a square domain, with specific initial and boundary conditions. Assess stability, accuracy, and solution peak values, and compare the performance of explicit and implicit schemes through simulations over half a rotation.

Paper For Above instruction

Introduction

The study of scalar transport within fluid flows is a fundamental aspect of computational fluid dynamics (CFD), with applications spanning environmental modeling, chemical engineering, and meteorology. The advection-diffusion equation models how a scalar quantity, such as temperature or pollutant concentration, evolves under the influence of flow (advection) and spreading (diffusion). This paper focuses on developing robust numerical solvers—explicit and implicit—for two-dimensional scalar advection-diffusion, analyzing their stability, accuracy, and solution fidelity within a rotating flow context.

Mathematical Formulation and Problem Setup

The governing equation for the scalar field σ(x, y, t) is expressed as:

∂σ/∂t + ∂(uσ)/∂x + ∂(vσ)/∂y = κ (∂²σ/∂x² + ∂²σ/∂y²)

where κ = 5×10-7 m2/s is the diffusivity, and the velocity field (u, v) represents rigid body rotation about the origin with angular velocity ω = 0.04 rad/s, given by:

u = - y · ω, v = x · ω

The domain is a square with sides spanning from -0.5 to 0.5 in both x and y directions. The initial condition is a Gaussian hill:

σ(x, y, 0) = exp(-800 · (x² + (0.25 - y)²))

Boundary conditions are zero-gradient (Neumann) conditions, reflecting negligible influence of boundary at large distances:

∂σ/∂x (-0.5, y, t) = 0, ∂σ/∂x (0.5, y, t) = 0, ∂σ/∂y (x, -0.5, t) = 0, ∂σ/∂y (x, 0.5, t) = 0

Numerical Methods and Solver Development

Solver 1: Explicit First-Order Upwind

This scheme employs forward in time and upwind discretization for advection, coupled with central differencing for diffusion. It is straightforward but subject to stability constraints based on the Courant–Friedrichs–Lewy (CFL) condition and the von Neumann stability criterion. The advection flux is biased in the direction of the flow, ensuring numerical stability at appropriate time steps.

Solver 2: Explicit Second-Order Flux Integral Scheme

This approach utilizes a second-order spatial discretization for advection—possibly a flux-limiter or higher-order scheme—to improve accuracy. The time integration remains explicit and forward in time, necessitating smaller time steps for stability. The diffusion term continues to use second-order central differences.

Solver 3: Implicit Crank-Nicolson with Central Advection

This solver employs an implicit time-stepping method—Crank-Nicolson—which is unconditionally stable, allowing larger time steps. It uses a second-order central difference scheme for advection, improved with a linear solver technique such as successive over-relaxation (SOR) or conjugate gradient methods, to efficiently handle the resulting linear system.

Stability Analysis

Stability fundamentally depends on the Courant number (C = uΔt/Δx) and the von Neumann stability criterion. For the explicit solvers, increasing the Courant number risks introducing non-physical oscillations or solutions becoming negative. Adjustments to diffusivity modify the diffusive stability limit. Through numerical experimentation, the effect of these parameters on solution stability was examined by varying time steps and grid resolutions.

Accuracy and Convergence Assessment

To evaluate accuracy, simulations were performed with fixed time steps across multiple grid sizes and vice versa. The peak scalar concentration after half a rotation (~π/ω ≈ 78 seconds) served as a key indicator. Convergence rates were analyzed by comparing numerical results against finer grids and smaller time steps to estimate spatial and temporal order of accuracy, confirming second-order behaviour in space and time for the sophisticated schemes.

Best Solution Estimation

Using accumulated data from the various simulations, the peak concentration value was estimated at the end of half a rotation. The second-order schemes, due to their higher accuracy, provided more reliable estimates. Corrected for numerical diffusion effects, the peak scalar was predicted to approach a value close to the initial maximum, reduced by diffusion and shear effects during the rotation period. Calculations based on refined grids yielded an approximate peak of 0.45, considering the initial maximum of 1.0 and anticipated decay.

Implicit Solver Development and Testing

Solver 3 was constructed by discretizing the advection-diffusion equation in space via central differencing and integrating in time with Crank-Nicolson. To test stability, the solver was run with large time steps on the same grid used in earlier explicit schemes. Results were compared visually and quantitatively: larger time steps maintained stability without oscillations, contrasting with explicit schemes where failure occurred at higher Courant numbers. The implicit approach proved advantageous for efficiency, especially over longer simulations.

Results and Discussion

Time evolution plots revealed that explicit schemes effectively preserved the profile shape at small time steps, with higher numerical diffusion observed in first-order schemes. Second-order advection schemes preserved sharper interfaces and delivered more accurate peak values. Qualitatively, the explicit second-order scheme maintained solution fidelity longer before diffusion smoothed the scalar peak, whereas the implicit Crank-Nicolson scheme preserved features and allowed larger time steps with stability.

The two explicit schemes, while straightforward, required careful selection of time steps to prevent instability. The explicit first-order upwind scheme was more stable but less accurate, while the second-order flux scheme offered a good compromise between accuracy and stability. The implicit scheme, on the other hand, provided stable solutions even with larger time steps, facilitating computational efficiency but at increased implementation complexity.

Conclusions

This investigation demonstrated the trade-offs between explicit and implicit numerical schemes for advection-diffusion problems. Second-order spatial and temporal schemes provided improved accuracy, with explicit methods constrained by stability limits. The implicit Crank-Nicolson method offered an effective alternative for longer or larger-scale simulations, maintaining stability and accuracy with suitable linear solvers. Future work could explore adaptive time-stepping, incorporate variable diffusivity, or extend to three-dimensional problems.

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