Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained

A HYBRID METHOD FOR STIFF REACTION-DIFFUSION EQUATIONS.

The second-order implicit integration factor method (IIF2) is effective at solving stiff reaction-diffusion equations owing to its nice stability condition. IIF has previously been applied primarily to systems in which the reaction contained no explicitly time-dependent terms and the boundary conditions were homogeneous. If applied to a system with explicitly time-dependent reaction terms, we find that IIF2 requires prohibitively small time-steps, that are relative to the square of spatial grid sizes, to attain its theoretical second-order temporal accuracy. Although the second-order implicit exponential time differencing (iETD2) method can accurately handle explicitly time-dependent reactions, it is more computationally expensive than IIF2. In this paper, we develop a hybrid approach that combines the advantages of both methods, applying IIF2 to reaction terms that are not explicitly time-dependent and applying iETD2 to those which are. The second-order hybrid IIF-ETD method (hIFE2) inherits the lower complexity of IIF2 and the ability to remain second-order accurate in time for large time-steps from iETD2. Also, it inherits the unconditional stability from IIF2 and iETD2 methods for dealing with the stiffness in reaction-diffusion systems. Through a transformation, hIFE2 can handle nonhomogeneous boundary conditions accurately and efficiently. In addition, this approach can be naturally combined with the compact and array representations of IIF and ETD for systems in higher spatial dimensions. Various numerical simulations containing linear and nonlinear reactions are presented to demonstrate the superior stability, accuracy, and efficiency of the new hIFE method

An exploration of the IGA method for efficient reservoir simulation

Novel numerical methods present exciting opportunities to improve the efficiency of reservoir simulators. Because potentially significant gains to computational speed and accuracy may be obtained, it is worthwhile explore alternative computational algorithms for both general and case-by-case application to the discretization of the equations of porous media flow, fluid-structure interaction, and/or production. In the present work, the fairly new concept of isogeometric analysis (IGA) is evaluated for its suitability to reservoir simulation via direct comparison with the industry standard finite difference (FD) method and 1st order standard finite element method (SFEM). To this end, two main studies are carried out to observe IGA’s performance with regards to geometrical modeling and ability to capture steep saturation fronts. The first study explores IGA’s ability to model complex reservoir geometries, observing L2 error convergence rates under a variety of refinement schemes. The numerical experimental setup includes an 'S' shaped line sink of varying curvature from which water is produced in a 2D homogenous domain. The accompanying study simplifies the domain to 1D, but adds in multiphase physics that traditionally introduce difficulties associated with modeling of a moving saturation front. Results overall demonstrate promise for the IGA method to be a particularly effective tool in handling geometrically difficult features while also managing typically challenging numerical phenomena.Petroleum and Geosystems Engineerin

A second order directional split exponential integrator for systems of advection–diffusion–reaction equations

We propose a second order exponential scheme suitable for two-component coupled systems of stiff evolutionary advection–diffusion–reaction equations in two and three space dimensions. It is based on a directional splitting of the involved matrix functions, which allows for a simple yet efficient implementation through the computation of small sized exponential-like functions and tensor-matrix products. The procedure straightforwardly extends to the case of an arbitrary number of components and to any space dimension. Several numerical examples in 2D and 3D with physically relevant (advective) Schnakenberg, FitzHugh–Nagumo, DIB, and advective Brusselator models clearly show the advantage of the approach against state-of-the-art techniques

Implicit time integration for high-order compressible flow solvers

The application of high-order spectral/hp element discontinuous Galerkin (DG) methods to unsteady compressible flow simulations has gained increasing popularity. However, the time step is seriously restricted when high-order methods are applied to an explicit solver. To eliminate this restriction, an implicit high-order compressible flow solver is developed using DG methods for spatial discretization, diagonally implicit Runge-Kutta methods for temporal discretization, and the Jacobian-free Newton-Krylov method as its nonlinear solver. To accelerate convergence, a block relaxed Jacobi preconditioner is partially matrix-free implementation with a hybrid calculation of analytical and numerical Jacobian.The problem of too many user-defined parameters within the implicit solver is then studied. A systematic framework of adaptive strategies is designed to relax the difficulty of parameter choices. The adaptive time-stepping strategy is based on the observation that in a fixed mesh simulation, when the total error is dominated by the spatial error, further decreasing of temporal error through decreasing the time step cannot help increase accuracy but only slow down the solver. Based on a similar error analysis, an adaptive Newton tolerance is proposed based on the idea that the iterative error should be smaller than the temporal error to guarantee temporal accuracy. An adaptive strategy to update the preconditioner based on the Krylov solver’s convergence state is also discussed. Finally, an adaptive implicit solver is developed that eliminates the need for repeated tests of tunning parameters, whose accuracy and efficiency are verified in various steady/unsteady simulations. An improved shock-capturing strategy is also proposed when the implicit solver is applied to high-speed simulations. Through comparisons among the forms of three popular artificial viscosities, we identify the importance of the density term and add density-related terms on the original bulk-stress based artificial viscosity. To stabilize the simulations involving strong shear layers, we design an extra shearstress based artificial viscosity. The new shock-capturing strategy helps dissipate oscillations at shocks but has negligible dissipation in smooth regions.Open Acces