P-adaptive Discontinuous Galerkin Method Research Articles

P-adaptive discontinuous Galerkin method for the shallow water equations with a parameter-free error indicator

We propose a p-adaptive quadrature-free discontinuous Galerkin method for the shallow water equations based on a computationally efficient adaptivity indicator which works without any problem-dependent parameters. The error and smoothness of the solution are detected using the information collected for slope limiting and, for piecewise constant discretizations, by carrying out a reconstruction procedure. The accuracy and robustness of the new scheme are evaluated using several benchmarks and compared to other adaptivity indicators. Our results indicate that the proposed indicator finds a good balance between solution quality and computational overhead.

High resolution viscous fingering simulation in miscible displacement using a p-adaptive discontinuous Galerkin method with algebraic multigrid preconditioner

High resolution simulation of viscous fingering can offer an accurate and detailed prediction for subsurface engineering processes involving fingering phenomena. The fully implicit discontinuous Galerkin (DG) method has been shown to be an accurate and stable method to model viscous fingering with high Peclet number and mobility ratio. In this paper, we present two techniques to speedup large scale simulations of this kind. The first technique relies on a simple p-adaptive scheme in which high order basis functions are employed only in elements near the finger fronts where the concentration has a sharp change. As a result, the number of degrees of freedom is significantly reduced and the simulation yields almost identical results to the more expensive simulation with uniform high order elements throughout the mesh. The second technique for speedup involves improving the solver efficiency. We present an algebraic multigrid (AMG) preconditioner which allows the DG matrix to leverage the robust AMG preconditioner designed for the continuous Galerkin (CG) finite element method. The resulting preconditioner works effectively for fixed order DG as well as p-adaptive DG problems. With the improvements provided by the p-adaptivity and AMG preconditioning, we can perform high resolution three-dimensional viscous fingering simulations required for miscible displacement with high Peclet number and mobility ratio in greater detail than before for well injection problems.

A P-adaptive Discontinuous Galerkin Method Using Local Time-stepping Strategy Applied to the Shallow Water Equations

A P-adaptive Discontinuous Galerkin Method Using Local Time-stepping Strategy Applied to the Shallow Water Equations

A semi-implicit, semi-Lagrangian, p-adaptive discontinuous Galerkin method for the shallow water equations

A semi-implicit and semi-Lagrangian discontinuous Galerkin method for the shallow water equations is proposed, for applications to geophysical scale flows. A non conservative formulation of the advection equation is employed, in order to achieve a more treatable form of the linear system to be solved at each time step. The method is equipped with a simple p-adaptivity criterion, that allows to adjust dynamically the number of local degrees of freedom employed to the local structure of the solution. Numerical results show that the method captures well the main features of gravity and inertial gravity waves, as well as reproducing correct solutions in nonlinear test cases with analytic solutions. The accuracy and effectiveness of the method are also demonstrated by numerical results obtained at high Courant numbers and with automatic choice of the local approximation degree.

A non-uniform basis order for the discontinuous Galerkin method of the acoustic and elastic wave equations

Here, we solve the time-dependent acoustic and elastic wave equations using the discontinuous Galerkin method for spatial discretization and the low-storage Runge–Kutta and Crank–Nicolson methods for time integration. The aim of the present paper is to study how to choose the order of polynomial basis functions for each element in the computational mesh to obtain a predetermined relative error. In this work, the formula 2 p + 1 ≈ κ h k , which connects the polynomial basis order p, mesh parameter h, wave number k, and free parameter κ, is studied. The aim is to obtain a simple selection method for the order of the basis functions so that a relatively constant error level of the solution can be achieved. The method is examined using numerical experiments. The results of the experiments indicate that this method is a promising approach for approximating the degree of the basis functions for an arbitrarily sized element. However, in certain model problems we show the failure of the proposed selection scheme. In such a case, the method provides an initial basis for a more general p-adaptive discontinuous Galerkin method.

A dynamic p-adaptive Discontinuous Galerkin method for viscous flow with shocks

A dynamic p-adaptive Discontinuous Galerkin method for compressible flows is proposed. The key element is a sensor, which measures the local regularity of the solution. This sensor is designed to preserve the compactness of the method and is easy to implement. In regions where the solution is quasi-uniform and in the vicinity of shocks the degree of the polynomial basis is decreased, while a high-degree basis is used in regions of smooth fluctuations of the flow. Numerical tests carried out in 1D and on 2D unstructured meshes prove that the convergence rate of the hybrid method is equal to the one of the highest degree polynomial basis, but with a significant CPU cost saving.