Computational Schemes for Initial-Boundary Value Problems. Many physical phenomena can be modelled as initial-boundary value problems described by partial differential equations. Simulations of such models require efficient and robust computational algorithms. The main aim of this project is to propose numerical algorithms for two dimensional spatial problems and three dimensional time-space models. A major focus of the project is to investigate methods that require about half the computation ....Computational Schemes for Initial-Boundary Value Problems. Many physical phenomena can be modelled as initial-boundary value problems described by partial differential equations. Simulations of such models require efficient and robust computational algorithms. The main aim of this project is to propose numerical algorithms for two dimensional spatial problems and three dimensional time-space models. A major focus of the project is to investigate methods that require about half the computational resources over celebrated schemes for solving boundary value problems.Read moreRead less
A new numerical analysis for partial differential equations with noise. This project aims to design novel numerical methods, grounded in rigorous mathematical foundations, for partial differential equations with stochastic source terms, such as for instance those modelling fluid flows with random perturbations. To ensure the accuracy of numerical simulations, preserving certain quantities of importance (mass, flux) is critical. The project's goal is to develop finite volume and high-order numeri ....A new numerical analysis for partial differential equations with noise. This project aims to design novel numerical methods, grounded in rigorous mathematical foundations, for partial differential equations with stochastic source terms, such as for instance those modelling fluid flows with random perturbations. To ensure the accuracy of numerical simulations, preserving certain quantities of importance (mass, flux) is critical. The project's goal is to develop finite volume and high-order numerical methods that are applicable in real-world settings, designed to achieve this preservation of essential quantities, and mathematically proven to be robust. The expected benefits are cost-efficient and reliable numerical tools for the scientific simulation of phenomena subjected to uncontrolled influence.Read moreRead less
Discrete functional analysis: Bridging pure and numerical mathematics. This project aims to create the first numerical analysis tools to design robust, mathematically proven algorithms for engineering problems in underground flows. These equations are essential to accurately model and understand phenomena such as oil extraction, carbon sequestration and groundwater contamination. The project will provide powerful mathematical tools to improve the reliability of numerical simulations for such cha ....Discrete functional analysis: Bridging pure and numerical mathematics. This project aims to create the first numerical analysis tools to design robust, mathematically proven algorithms for engineering problems in underground flows. These equations are essential to accurately model and understand phenomena such as oil extraction, carbon sequestration and groundwater contamination. The project will provide powerful mathematical tools to improve the reliability of numerical simulations for such challenges and significantly improve the reliability of the predictions under assumptions that are compatible with field applications.Read moreRead less
GEOMETRIC NUMERICAL INTEGRATION. Many scientific phenomena in physics, astronomy, and chemistry, are modelled by ordinary differential equations (ODEs). Often these equations have no solution in closed form, and one relies on numerical integration. Traditionally this is done using Runge-Kutta methods or linear multistep methods. In the last decade, however, we (and others) have discovered novel classes of so-called "geometric" numerical integration methods that preserve qualititative featur ....GEOMETRIC NUMERICAL INTEGRATION. Many scientific phenomena in physics, astronomy, and chemistry, are modelled by ordinary differential equations (ODEs). Often these equations have no solution in closed form, and one relies on numerical integration. Traditionally this is done using Runge-Kutta methods or linear multistep methods. In the last decade, however, we (and others) have discovered novel classes of so-called "geometric" numerical integration methods that preserve qualititative features of certain ODE's exactly (in contrast to traditional methods), leading to crucial stability improvements. Extending concepts from dynamical systems theory and traditional numerical ODEs, this project will improve, extend and systematize this new field of geometric integration.Read moreRead less
Geometric Integration. This project gives an important boost to Australia's strength in the niche area of geometric numerical integration,in the face of strong international competition. It gathers 7 world experts from 5 countries to create new computer programs to improve calculations in dynamics, with applications ranging from astronomy, physics, chemistry, biology, and meteorology to finance. It strengthens Australia's links with the mathematical software industry, and will lead to world-clas ....Geometric Integration. This project gives an important boost to Australia's strength in the niche area of geometric numerical integration,in the face of strong international competition. It gathers 7 world experts from 5 countries to create new computer programs to improve calculations in dynamics, with applications ranging from astronomy, physics, chemistry, biology, and meteorology to finance. It strengthens Australia's links with the mathematical software industry, and will lead to world-class graduates and research training.Read moreRead less
Geometric numerical integration of differential equations. Differential equations (DEs) play a central role in modelling scientific phenomena in physics, biology, chemistry, astronomy, meteorology, and geoscience. We have developed new ways of solving DEs, using geometric integration, which have significant advantages over traditional methods because of the crucial nonlinear stability they provide.
This project, combining 7 world experts from 6 countries and 1 early career researcher, will pl ....Geometric numerical integration of differential equations. Differential equations (DEs) play a central role in modelling scientific phenomena in physics, biology, chemistry, astronomy, meteorology, and geoscience. We have developed new ways of solving DEs, using geometric integration, which have significant advantages over traditional methods because of the crucial nonlinear stability they provide.
This project, combining 7 world experts from 6 countries and 1 early career researcher, will place Australia at the forefront of this intensive international activity.
It will significantly strengthen Australia's links with the mathematical software industry (e.g. Wolfram Research, Inc), and will lead to world class graduates and research training.
Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE130100333
Funder
Australian Research Council
Funding Amount
$315,640.00
Summary
A new class of fast and reliable spectral methods for partial differential equations. The project will develop novel fast and reliable spectral methods for computing solutions to general partial differential equations. These methods will be applied to solve important equations that arise in mathematical physics and other areas where high accuracy is essential.
Novel mathematics and numerical methods for ferromagnetic problems. This project aims to develop novel mathematical theories and numerical methods for ferromagnetic problems. These problems arise from many real-life applications, for example in storage devices and magnetic sensors, which are often affected by random (thermal) noise. Since thermal noise limits the data-retention time of the devices, analysing the effect of noise is highly significant. Expected outcomes will be novel computational ....Novel mathematics and numerical methods for ferromagnetic problems. This project aims to develop novel mathematical theories and numerical methods for ferromagnetic problems. These problems arise from many real-life applications, for example in storage devices and magnetic sensors, which are often affected by random (thermal) noise. Since thermal noise limits the data-retention time of the devices, analysing the effect of noise is highly significant. Expected outcomes will be novel computational techniques to solve the underlying equations and deal with randomness. The project aims to put Australia in the forefront of international research in numerical methods in micromagnetism. The new computational methods are expected to be used to advance technology in magnetic memory devices.Read moreRead less
Interface-aware numerical methods for stochastic inverse problems. This project aims to design novel high-performance numerical tools for solving large-scale forward and inverse problems dominated by stochastic interfaces and quantifying associated uncertainties. In real-world applications such as groundwater, these tools are instrumental for assimilating big datasets into mathematical models for providing reliable predictions. By advancing and integrating high-order polytopal schemes, multileve ....Interface-aware numerical methods for stochastic inverse problems. This project aims to design novel high-performance numerical tools for solving large-scale forward and inverse problems dominated by stochastic interfaces and quantifying associated uncertainties. In real-world applications such as groundwater, these tools are instrumental for assimilating big datasets into mathematical models for providing reliable predictions. By advancing and integrating high-order polytopal schemes, multilevel methods, transport maps, and dimension reduction, this project's anticipated outcomes are highly accurate and cost-efficient numerical schemes, certified by rigorous mathematical analysis. This should provide data-centric simulation tools with enhanced reliability, for engineering and scientific applications.Read moreRead less
Towards predictive 4D computational models for the heart. This project aims to develop novel high-performance numerical algorithms for multiscale and multiphysics PDEs with dynamic interfaces, the development and analysis of a novel PDE system modelling the electromechanics of heart and torso, and the combination of these numerical techniques and models to deliver predictive tools for patient-specific simulations of the cardiac function. It involves the design and mathematical analysis of space- ....Towards predictive 4D computational models for the heart. This project aims to develop novel high-performance numerical algorithms for multiscale and multiphysics PDEs with dynamic interfaces, the development and analysis of a novel PDE system modelling the electromechanics of heart and torso, and the combination of these numerical techniques and models to deliver predictive tools for patient-specific simulations of the cardiac function. It involves the design and mathematical analysis of space-time variational discretisations on embedded meshes, 4D computational geometry algorithms for numerical integration and multilevel solvers. By combining scientific computing and machine learning, one anticipated outcome of this research is a new generation of nonlinear PDE approximations and solvers.Read moreRead less