# A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

## TL;DR

This paper introduces a regularized discrete least squares collocation method combined with finite volume techniques for solving complex two-dimensional nonlinear time-dependent PDEs on irregular domains, emphasizing high accuracy and computational stability.

## Contribution

The paper presents a novel mesh-free finite volume method that reduces to a collocation method as the control volume radius approaches zero, with regularization improving system conditioning.

## Key findings

- The methods achieve high spatial accuracy on irregular domains.
- Regularization leads to well-conditioned systems solvable by QR factorization.
- The approach effectively handles coupled time-fractional PDE systems.

## Abstract

In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain.