Convergent numerical methods for nonlinear partial differential equations with applications in complex fluids and tumor modeling

Partial differential equations (PDEs) are a powerful tool to describe a variety of phenomena in the sciences and engineering applications in a mathematical way. However, as the models are typically quite complex, solutions to the PDEs cannot be computed explicitly, and numerical algorithms are needed to compute approximations to the solutions using computers. In this project we design numerical methods for mathematical models for complex fluids and tumor growth models. Complex fluids are materials with unusual responses to applied stress or strain. This makes them very interesting for applications in biotechnology, medicine, chemistry and others. So far we have focused on the analysis and simulation of partial differential equations modeling liquid crystals and ferrofluids as part of this project. Cancer is among the leading causes of death worldwide. Despite lots of research efforts and the development of new therapies and technologies, many forms of cancer are still incurable. The hope is that mathematical models for tumor progression can contribute to understanding cancer better. As part of this project, we have recently designed a numerical method that can simulate features of tumors that appear in real tumors. Recently, we have extended our research to investigating the mathematical foundations of the turbulence theory developed by Kolmogorov in 1941.

We made a tiny contribution to the mathematical theory of turbulence and attempted to understand tumor growth and ferrofluids better via mathematical and numerical analysis. Understanding turbulence better may lead to safer and more energy-efficient air travel, improved weather predictions, and assessment of natural hazards. The mathematical and computational study of mathematical models of tumor growth and ferrofluids may lead to a better understanding of these phenomena and aid in the development of computational tools for prediction of cancer progression and ferrofluid dynamics.

Whereas the theory and approximation of linear partial differential equations is relatively well-developed, there are still many open questions as far as nonlinear PDEs are concerned. Solutions to such equations often exhibit complex structures. They may develop shock waves, rapid oscillations, and blow-ups. For this reason, they are not differentiable in the classical sense, and weaker, more general notions of differentiability and solution need to be defined. Proving well-posedness of such equations often turns out to be a challenging task, as the most famous example we mention the problem of proving smoothness of the Navier-Stokes equations in three space dimensions. This also complicates the task of developing stable and convergent numerical methods to approximate nonlinear PDEs. Already the design of a stable method can sometimes prove to be a challenge and proving convergence requires discretized versions of the techniques used at the continuous level of the PDE or alternative techniques which poses additional difficulty. For this reason, there is a lack of convergent, but still practically applicable numerical methods for many types of nonlinear PDEs. This is particularly the case in the field of complex fluids and mathematical modeling of tumor growth. An abundance of mathematical models involving nonlinear PDEs is available to describe these phenomena. However, since the models are in general very complicated, few convergent numerical methods exist at the present time, or in cases where numerical studies have been conducted, they only apply to simplified models with restrictive assumptions. The aim of this research project is to develop computationally efficient and convergent numerical methods for nonlinear PDEs that model realistic scenarios in complex fluids and tumor modeling which will subsequently allow us to compare their results with medical data and physical experiments for further predictions on cancer progression and the dynamics of polymers.