Isogeometric Residual Minimization Method (iGRM) with Direction Splitting for Non-Stationary Advection-Diffusion Problems

TL;DR

This paper introduces iGRM with direction splitting, a novel implicit method combining isogeometric analysis, residual minimization, and alternating direction solver for efficient non-stationary advection-diffusion simulations.

Contribution

The paper presents a new implicit computational method that leverages tensor product B-splines, residual minimization, and Kronecker structure for efficient solving of advection-diffusion problems.

Findings

Achieves linear computational cost with implicit schemes.

Successfully applied to complex advection-diffusion examples.

Demonstrates stability and efficiency of the proposed method.

Abstract

In this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, implicit dynamics, residual minimization, and alternating direction solver. We utilize tensor product B-spline basis functions in space, implicit second order time integration schemes, residual minimization in every time step, and we exploit Kronecker product structure of the matrix to employ linear computational cost alternating direction solver. We implement an implicit time integration scheme and apply, for each space-direction, a stabilized mixed method based on residual minimization. We show that the resulting system of linear equations has a Kronecker product structure, which results in a linear computational cost of the direct solver, even using implicit time integration schemes together with the stabilized mixed formulation. We test our method on three advection-diffusion computational examples, including model ``membrane'' problem, the circular wind problem, and the simulations modeling pollution propagating from a chimney.