Related papers: Holistic finite differences ensure fidelity to Bur…
We analyse the nonlinear Kuramoto-Sivashinsky equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately…
I prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small. Using such tools of dynamical systems theory gives new assurances about the quality of…
Modern dynamical systems theory has previously had little to say about finite difference and finite element approximations of partial differential equations (Archilla, 1998). However, recently I have shown one way that centre manifold…
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate,…
We analyse the nonlinear Kuramoto--Sivashinsky equation to develop accurate discretisations modeling its dynamics on coarse grids. The analysis is based upon centre manifold theory so we are assured that the discretisation accurately models…
Coarse grained, macroscale, spatial discretisations of nonlinear nonautonomous partial differential\difference equations are given novel support by centre manifold theory. Dividing the physical domain into overlapping macroscale elements…
I previously used Burgers' equation to introduce a new method of numerical discretisation of \pde{}s. The analysis is based upon centre manifold theory so we are assured that the discretisation accurately models all the processes and their…
Constructing discrete models of stochastic partial differential equations is very delicate. Stochastic centre manifold theory provides novel support for coarse grained, macroscale, spatial discretisations of nonlinear stochastic partial…
I introduce an innovative methodology for deriving numerical models of systems of partial differential equations which exhibit the evolution of spatial patterns. The new approach directly produces a discretisation for the evolution of the…
Many practical approximations in physics and engineering invoke a relatively long physical domain with a relatively thin cross-section. In this scenario we typically expect the system to have structures that vary slowly in the long…
Burgers equation is one of the simplest nonlinear partial differential equations-it combines the basic processes of diffusion and nonlinear steepening. In some applications it is appropriate for the diffusion coefficient to be a…
Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is…
Burgers' equation is a well-studied model in applied mathematics with connections to the Navier-Stokes equations in one spatial direction and traffic flow, for example. Following on from previous work, we analyse solutions to Burgers'…
Constructing numerical models of noisy partial differential equations is very delicate. Our long term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a…
The long term aim is to use modern dynamical systems theory to derive discretisations of noisy, dissipative partial differential equations. As a first step we here consider a small domain and apply stochastic centre manifold techniques to…
Many physical systems are well described on domains which are relatively large in some directions but relatively thin in other directions. In this scenario we typically expect the system to have emergent structures that vary slowly over the…
Developments in dynamical systems theory provides new support for the macroscale modelling of pdes and other microscale systems such as Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically resolving subgrid…
Data-driven turbulence modeling is experiencing a surge in interest following algorithmic and hardware developments in the data sciences. We discuss an approach using the differentiable physics paradigm that combines known physics with…
Complex systems may be subject to various uncertainties. A great effort has been concentrated on predicting the dynamics under uncertainty in initial conditions. In the present work, we consider the well-known Burgers equation with random…
In this article we study generalizations of the inhomogeneous Burgers equation. First at the operator level, in the sense that we replace classical differential derivations by operators with certain properties, and then we increase the…