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Register a new userSmooth subgrid fields underpin rigorous closure in spatial discretisation of reaction-advection-diffusion PDEs
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Finite difference/element/volume methods of discretising PDEs impose a subgrid scale interpolation on the dynamics. In contrast, the holistic discretisation approach developed herein constructs a natural subgrid scale field adapted to the whole system out-of-equilibrium dynamics. Consequently, the macroscale discretisation is fully informed by the underlying microscale dynamics. We establish a new proof that in principle there exists an exact closure of the dynamics of a general class of reaction-advection-diffusion PDEs, and show how our approach constructs new systematic approximations to the in-principle closure starting from a simple, piecewise-linear, continuous approximation. Under inter-element coupling conditions that guarantee continuity of several field properties, the holistic discretisation possesses desirable properties such as a natural cubic spline first-order approximation to the field, and the self-adjointness of the diffusion operator under periodic, Dirichlet and Neumann macroscale boundary conditions. As a concrete example, we demonstrate the holistic discretisation procedure on the well-known Burgers PDE, and compare the theoretical and numerical stability of the resulting discretisation to other approximations. The approach developed here promises to be able to systematically construct automatically good, macroscale discretisations to a wide range of PDEs, including wave PDEs.
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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, macroscale closures of discrete models of the pde. Here we explore reaction-diffusion problems in two spatial dimensions. Centre manifold theory ensures that slow manifold, holistic, discretisations exists, are quickly attractive, and are systematically approximated. Special coupling of the finite elements ensures that the resultant discretisations are consistent with the pde to as high an order as desired. Computer algebra handles the enormous algebraic details as seen in the specific application to the Ginzburg--Landau equation. However, higher order models in 2D appear to require a mixed numerical and algebraic approach that is also developed. Being driven by the residuals of the equations, the modelling here may be straightforwardly adapted to a wide class of reaction-diffusion differential and lattice equations in multiple space dimensions.
A study is presented on the convergence of the computation of coupled advection-diffusion-reaction equations. In the computation, the equations with different coefficients and even types are assigned in two subdomains, and Schwarz iteration is made between the equations when marching from a time level to the next one. The analysis starts with the linear systems resulting from the full discretization of the equations by explicit schemes. Conditions for convergence are derived, and its speedup and the effects of difference in the equations are discussed. Then, it proceeds to an implicit scheme, and a recursive expression for convergence speed is derived. An optimal interface condition for the Schwarz iteration is obtained, and it leads to perfect convergence, that is, convergence within two times of iteration. Furthermore, the methods and analyses are extended to the coupling of the viscous Burgers equations. Numerical experiments indicate that the conclusions, such as the perfect convergence, drawn in the linear situations may remain in the Burgers equations computation.
We provide a preliminary comparison of the dispersion properties, specifically the time-amplification factor, the scaled group velocity and the error in the phase speed of four spatiotemporal discretization schemes utilized for solving the one-dimensional (1D) linear advection diffusion reaction (ADR) equation: (a) An explicit (RK2) temporal integration combined with the Optimal Upwind Compact Scheme (or OUCS3) and the central difference scheme (CD2) for second order spatial discretization, (b) a fully implicit mid-point rule for time integration coupled with the OUCS3 and the Leles compact scheme for first and second order spatial discretization, respectively, (c) An implicit (mid-point rule)-explicit (RK2) or IMEX time integration blended with OUCS3 and Leles compact scheme (where the IMEX time integration follows the same ideology as introduced by Ascher et al.), and (d) the IMEX (mid-point/RK2) time integration melded with the New Combined Compact Difference scheme (or NCCD scheme). Analysis reveal the superior resolution features of the IMEX-NCCD scheme including an enhanced region of neutral stability (a region where the amplification factor is close to one), a diminished region of spurious propagation characteristics (or a region of negative group velocity) and a smaller region of nonzero phase speed error. The dispersion error of these numerical schemes through the role of q-waves is further investigated using the novel error propagation equation for the 1D linear ADR equation. Again, the in silico experiments divulge excellent Dispersion Relation Preservation (DRP) properties of the IMEX-NCCD scheme including minimal dissipation via implicit filtering and negligible unphysical oscillations (or Gibbs phenomena) on coarser grids.
We analyse a PDE system modelling poromechanical processes (formulated in mixed form using the solid deformation, fluid pressure, and total pressure) interacting with diffusing and reacting solutes in the medium. We investigate the well-posedness of the nonlinear set of equations using fixed-point theory, Fredholms alternative, a priori estimates, and compactness arguments. We also propose a mixed finite element method and rigorously demonstrate the stability of the scheme. Error estimates are derived in suitable norms, and numerical experiments are conducted to illustrate the mechano-chemical coupling and to verify the theoretical rates of convergence.
We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented.
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