Asynchronous and exponential based numerical schemes for porous media flow
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A great many physical phenomena are modelled by partial di erential equations (PDEs), and numerical schemes often have to be employed to approximate the solutions to these equations where analytical solutions cannot be found. We develop and analyse here new schemes belonging to two broad classes, schemes that are asynchronous, and exponential integrators. We apply these schemes to test models of advection-di usion-reaction processes that occur in porous media ow. Asynchronous schemes allow di erent parts of the physical domain to evolve at different rates. We develop a class of asynchronous schemes that progress by discrete events, where a single event is the transfer of a unit of mass through the domain, according to the local ux. These schemes are intended to focus computational e ort where it is most needed, as a high local ux will cause the algorithm to automatically take more events in that part of the domain. We develop the simplest version of this scheme, and then develop further schemes by adding modi cations to address potential shortcomings. Numerical experiments indicate a number of interesting relations between the parameters of these schemes. Particularly, the error of the schemes seems to be rst order with respect to a control parameter we call the mass unit. Some analysis is conducted which can pave the way towards robust theoretical understanding of these schemes in the future. Exponential integrators are time stepping schemes which exactly solve the linear part of a semilinear ODE system. This class of schemes requires the approximation of a matrix exponential in every step, and one successful modern method is the Krylov subspace projection method. We investigate, through analysis and experiment, the e ect of breaking down a single timestep into multiple substeps, recycling the Krylov subspace to minimise costs. Our results indicate that this can increase accuracy and e ciency. We show the results of an investigation into developing a class of `semi-exponential' Runge-Kutta type schemes, which use an exponential integrator for the initial stage and then essentially ful l classical order conditions for the remaining stages. Finally, we return to the concept of asynchronicity in a di erent form. With the advent of massively parallel machines, there is increasing interest in developing domain-decomposition type schemes that are robust to random failures or delays in communication between processing elements. This is because in massively parallel machines, communication between processors is likely to be the signi cant bottleneck in execution time. Recently the e ect of such communication delay with a simple domain-decomposed Euler timestepping solver applied to a linear PDE has been investigated with promising results. Here, inspired by exponential integrators, we investigate the natural extension of this, by replacing the Euler timestepping with the evaluation of the appropriate matrix exponential on the sub-domain. We have performed experiments simulating the communication delay and the results are also promising.