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
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.
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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
Discovery Early Career Researcher Award - Grant ID: DE170100222
Funder
Australian Research Council
Funding Amount
$313,964.00
Summary
Optimal adaptivity for uncertainty quantification. This project aims to use an adaptive mesh refinement algorithm to improve the ratio of approximation accuracy versus computational time. Partial differential equations with random coefficients are crucial in simulating groundwater flow, structural stability and composite materials, but their numerical approximation is difficult and time consuming. Advances in adaptive mesh refinement theory allow full analysis and mathematical understanding of t ....Optimal adaptivity for uncertainty quantification. This project aims to use an adaptive mesh refinement algorithm to improve the ratio of approximation accuracy versus computational time. Partial differential equations with random coefficients are crucial in simulating groundwater flow, structural stability and composite materials, but their numerical approximation is difficult and time consuming. Advances in adaptive mesh refinement theory allow full analysis and mathematical understanding of the convergence behaviour of the proposed algorithm. The project intends to develop a theory of adaptive algorithms and freely available software for their reliable (and mathematically underpinned) simulation which could solve problems beyond the capabilities of even the most powerful computers.Read moreRead less
Multiscale modelling of systems with complex microscale detail. This project aims to develop systematic mathematical and computational methods for the compact and accurate macroscale modelling of systems with microscopic irregular details. The methodology, justified with mathematical analysis and computation, uses small bursts of particle/agent simulations, partial differential equation (PDEs), or difference equations, to efficiently predict macroscale behaviour. This project’s mathematical meth ....Multiscale modelling of systems with complex microscale detail. This project aims to develop systematic mathematical and computational methods for the compact and accurate macroscale modelling of systems with microscopic irregular details. The methodology, justified with mathematical analysis and computation, uses small bursts of particle/agent simulations, partial differential equation (PDEs), or difference equations, to efficiently predict macroscale behaviour. This project’s mathematical methodology aims to efficiently and accurately extract and simulate the collective dynamics which emerge on macroscales, leading to improved prediction and understanding of the significant features of these complex systems at the scale relevant to engineers and scientists.Read moreRead less
Quantifying uncertainty: innovations in high dimensional computation. High dimensional problems (problems in which there are hundreds or thousands of continuous variables) arise in many applications, from ground water flow to mathematical physics and finance. They typically present major challenges to computational resources and serious mathematical challenges in devising new and improved methods and in proving that they are effective. The aim of this project is to develop new computational meth ....Quantifying uncertainty: innovations in high dimensional computation. High dimensional problems (problems in which there are hundreds or thousands of continuous variables) arise in many applications, from ground water flow to mathematical physics and finance. They typically present major challenges to computational resources and serious mathematical challenges in devising new and improved methods and in proving that they are effective. The aim of this project is to develop new computational methods and theory for high dimensional problems, and to apply these methods to significant applications. The results are expected to allow faster and more accurate solution of problems of growing importance.Read moreRead less