Accurate simulation of waves and sharp fronts
Conservation laws describe how quantities such as mass, momentum and energy are transported and conserved. Including sources and other forcing gives rise to balance laws. These equations underpin many flow and transport models, and their solutions can develop shocks and sharp fronts even from smooth initial conditions.
Numerical methods must resolve these features without smearing them through numerical diffusion or introducing spurious oscillations. High-resolution methods adapt the numerical approximation to the local solution, combining high accuracy in smooth regions with control of unwanted oscillations near discontinuities. This can improve results without substantially refining the entire grid.
How we can help
- Develop and select numerical methods: Adapt discretisations to the equations, grid and requirements for accuracy, robustness and computational cost.
- Improve existing simulators: Reduce numerical diffusion and spurious oscillations, and investigate how flux evaluation, reconstruction and time stepping affect the results.
- Preserve important physical properties: Design methods that maintain local conservation and relevant equilibrium states, and respect physical bounds such as non-negative water depth.
- Develop efficient implementations: Adapt algorithms and data structures for GPUs and parallel computing.
- Verify accuracy and performance: Compare methods using test problems and representative models, assessing errors, robustness and overall computational cost.
Our numerical expertise
We have extensive experience with high-resolution methods for hyperbolic conservation and balance laws. Our expertise includes:
- Godunov schemes and exact and approximate Riemann solvers.
- Total variation diminishing (TVD) schemes, slope limiters and flux limiters.
- Weighted essentially non-oscillatory (WENO) schemes for high-order reconstruction with control of oscillations near discontinuities.
- Central schemes, particularly semi-discrete central-upwind schemes.
- Discontinuous Galerkin methods.
- Front-tracking methods, which explicitly track discontinuities.
- Well-balanced discretisations that preserve selected equilibrium states by balancing flux and source terms.
Selecting a method involves more than its formal order of accuracy. We also consider how it handles sharp fronts, source terms, boundary conditions and grid geometry. For balance laws, it is particularly important that the discretisation does not introduce artificial motion into a physical equilibrium, such as water at rest over uneven bathymetry.
Applications in water flows and porous media
Much of our work concerns the shallow-water equations for surface water and urban flooding and coastal circulation and storm surges. These applications require methods that handle wave propagation, variations in bathymetry and terrain, and transitions between wet and dry areas.
We also work on transport equations for multiphase, multicomponent flow in porous media, including discretisations on complex polygonal and polyhedral grids. Our research experience also includes gas dynamics, traffic flow and magnetohydrodynamics.
From numerical methods to efficient software
We have more than two decades of experience implementing high-resolution methods on GPUs. Many of the computations are local and well suited to parallel execution, but good performance requires numerical methods, memory use and computational organisation to be considered together.
This experience is reflected in GPU Ocean, which supports ensembles of simplified ocean models, and Alsvinn, a parallel finite-volume simulator for conservation laws with support for uncertainty quantification.
Work with us
Do you need more accurate transport calculations, better resolution of sharp fronts or faster simulations? Contact our Applied Computational Science research group to discuss collaboration on numerical methods and simulator development.