[Paper Review] Numerical simulation of solutions and moments of the smoluchowski coagulation equation

Dustin D. Keck, David M. Bortz|arXiv (Cornell University)|Dec 27, 2013

Advanced Mathematical Modeling in Engineering4 citations

TL;DR

This paper compares finite element (FEM) and finite volume (FLFM) methods for numerically solving the Smoluchowski coagulation equation, focusing on accuracy in approximating solution moments and computational cost. It finds that FEM offers superior accuracy for slowly aggregating systems with significantly lower computational cost, while FLFM is slightly more accurate for the zeroth moment in fast aggregation and consistently more accurate for the first moment, with both methods showing second-order convergence.

ABSTRACT

Researchers have employed variations of the Smoluchowski coagulation equation to model a wide variety of both organic and inorganic phenomena and with relatively few known analytical solutions, numerical solutions play an important role in studying this equation. In this article, we consider numerical approximations, focusing on how different discretization schemes impact the accuracy of approximate solution moments. Pursuing the eventual goal of comparing simulated solutions to experimental data, we must carefully choose the numerical method most appropriate to the type of data we attain. Within this context, we compare and contrast the accuracy and computational cost of a finite element approach and a finite volume-based scheme. Our study provides theoretical and numerical evidence that the finite element approach achieves much more accuracy when the system aggregates slowly, and it does so with much less computation cost. Conversely, the finite volume method is slightly more accurate approximating the zeroth moment when the system aggregates quickly and is much more accurate approximating the first moment in general. Lastly, our study also provides numerical evidence that the finite element method (conventionally considered first order) actually belongs to a class of discontinuous Galerkin methods that exhibit superconvergence, or second order in our case.

Motivation & Objective

• To compare the accuracy and computational cost of finite element (FEM) and finite volume (FLFM) schemes for solving the Smoluchowski coagulation equation.
• To evaluate how discretization schemes affect the approximation of solution moments, particularly the zeroth and first moments, under varying aggregation dynamics.
• To investigate the impact of truncation parameter $x_{\text{max}}$ and grid spacing $\Delta x$ on numerical accuracy for both methods.
• To guide method selection based on experimental data type, such as Coulter counter (zeroth moment) or flow cytometry (first moment) data.
• To provide theoretical and numerical evidence on convergence rates and superconvergence behavior of the FEM.

Proposed method

• The study employs a finite element method (FEM) based on prior work by Banks and Kappel, extended by Ackleh and Fitzpatrick, for spatial discretization of the coagulation equation.
• A finite volume-type scheme, the Filbet and Laurençot Flux Method (FLFM), is used as a benchmark for comparison, known for second-order convergence in $L^1$.
• Numerical solutions are computed for two aggregation kernels: constant kernel $K_A(x,y) \equiv 1$ and multiplicative kernel $K_A(x,y) = xy$.
• The accuracy of the FEM and FLFM is evaluated by comparing their approximations to fine-grid reference solutions and to known moments.
• Grid refinement studies are conducted with varying $\Delta x$ and $x_{\text{max}}$ to assess sensitivity and convergence behavior.
• Moment approximations are computed via $M_i(f; x_1,x_2) = \int_{x_1}^{x_2} x^i f(t,x) \, dx$, with $f$ representing the size distribution and $g = x f$ the volume distribution.

Experimental results

Research questions

• RQ1 How do the FEM and FLFM compare in accuracy when approximating the full solution of the Smoluchowski coagulation equation?
• RQ2 Which method provides more accurate approximations of the zeroth and first moments under slow and fast aggregation dynamics?
• RQ3 What is the impact of the truncation parameter $x_{\text{max}}$ on the accuracy of each method, especially when experimental data is limited to small volume ranges?
• RQ4 Does the FEM exhibit superconvergence, as suggested by numerical evidence, despite being conventionally considered first-order?
• RQ5 For a given accuracy level, which method offers greater computational efficiency?

Key findings

• The FEM achieves second-order convergence in $L^1[\mathbf{X}]$, supporting the conjecture that it exhibits superconvergence, contrary to its conventional classification as first-order.
• For slowly aggregating systems ($K_A(x,y) \equiv 1$), the FEM achieves comparable solution accuracy to FLFM but with significantly lower computational cost.
• In slow aggregation, the FEM approximates the zeroth moment more accurately than FLFM, while FLFM is slightly more accurate for the first moment.
• For fast aggregation ($K_A(x,y) = xy$), the FLFM is more accurate in approximating the first moment across all cases, and slightly more accurate for the zeroth moment.
• The FEM is less sensitive to the choice of $x_{\text{max}}$ than the FLFM, especially under the multiplicative kernel, making it more robust when experimental data has limited upper detection limits.
• For equivalent accuracy, the FEM on 800 grid points offers substantial computational cost savings over the FLFM on 100 grid points.