Second-order Runge-Kutta Discontinuous Galerkin Research Articles

A discontinuous Galerkin approach for conservative modeling of fully nonlinear and weakly dispersive wave transformations

This work extends a robust second-order Runge–Kutta Discontinuous Galerkin (RKDG2) method to solve the fully nonlinear and weakly dispersive flows, within a scope to simultaneously address accuracy, conservativeness, cost-efficiency and practical needs. The mathematical model governing such flows is based on a variant form of the Green–Naghdi (GN) equations decomposed as a hyperbolic shallow water system with an elliptic source term. Practical features of relevance (i.e. conservative modeling over irregular terrain with wetting and drying and local slope limiting) have been restored from an RKDG2 solver to the Nonlinear Shallow Water (NSW) equations, alongside new considerations to integrate elliptic source terms (i.e. via a fourth-order local discretization of the topography) and to enable local capturing of breaking waves (i.e. via adding a detector for switching off the dispersive terms). Numerical results are presented, demonstrating the overall capability of the proposed approach in achieving realistic prediction of nearshore wave processes involving both nonlinearity and dispersion effects within a single model.

RKDG2 shallow-water solver on non-uniform grids with local time steps: Application to 1D and 2D hydrodynamics

This paper investigates local time stepping (LTS) with the RKDG2 (second-order Runge–Kutta Discontinuous Galerkin) non-uniform solutions of the inhomogeneous SWEs (shallow water equations) with source terms. A LTS algorithm – recently designed for homogenous hyperbolic PDE(s) – is herein reconsidered and improved in combination with the RKDG2 shallow-flow solver (LTS-RKDG2) including topography and friction source terms as well as wetting and drying. Two LTS-RKDG2 schemes that adapt 3 and 4 levels of LTSs are configured on 1D and/or 2D (quadrilateral) non-uniform meshes that, respectively, adopt 3 and 4 scales of spatial discretization. Selected shallow water benchmark tests are used to verify, assess and compare the LTS-RKDG2 schemes relative to their conventional Global Time Step RKDG2 alternatives (GTS-RKDG2) considering several issues of practical relevance to hydraulic modelling. Results show that the LTS-RKDG2 models could offer (depending on both the mesh setting and the features of the flow) comparable accuracy to the associated GTS-RKDG2 models with a savings in runtime of up to a factor of 2.5 in 1D simulations and 1.6 in 2D simulations.

Fully Coupled Discontinuous Galerkin Modeling of Dam-Break Flows over Movable Bed with Sediment Transport

A one-dimensional (1D) discontinuous Galerkin morphodynamic model has been devised with application to simulate of dam-break flows over erodible beds with suspended sediment transport. The morphodynamic equations adopt the shallow-water equations (SWE) considering the interaction of sediment transport and bed changes on the flow. A local second-order Runge-Kutta discontinuous Galerkin (RKDG2) model has been reformulated to numerically solve the morphodynamic equations in a fully coupled manner and with a noncapacity sediment model. The model’s implementation is thoroughly detailed with focus on the discretization of the complex source terms, the treatment of wetting and drying, and other stabilizing issues pertaining to high-solution gradients and the transient character of the topography. The model has been favorably applied to replicate experimental dam-break flow over erodible sediment beds.

Influence of Total-Variation-Diminishing Slope Limiting on Local Discontinuous Galerkin Solutions of the Shallow Water Equations

Finite volume (FV) slope limiting is essential to stabilize discontinuous Galerkin (DG) solutions despite a number of side effects such as local loss of accuracy and increased run-time cost. These side effects have been experienced with DG solutions to the homogeneous system of conservation laws and are usually accepted as long as they do not affect the reliability of the numerical predictions and provide a better stability property. They have also led to the concept of localizing the slope-limiting process. When a model is applied to simulate the flow problems that necessitate the solution of the inhomogeneous shallow water equations (SWEs) with/without flooding and drying, the slope-limiting routine can have extra adverse effects on the local DG framework. These effects, causes, and implications on DG solutions to the SWEs have not yet received adequate attention, and a full investigation is therefore needed. With the aim of improving a second-order Runge-Kutta discontinuous Galerkin (RKDG2) SWE solver, the effects of FV slope limiting on the RKDG2 flow solutions were analyzed. Three versions of the RKDG2 scheme were configured: the first and second versions were respectively associated with the local limiting (LL) and global limiting (GL) processes, which both use limiter-free linear front tracking (LFT) to trace a wet/dry front, and the third was implemented with the local limiting process but with constant front-tracking (CFT) of the wet/dry front. Selected hypothetical and practical hydraulic tests were investigated to demonstrate the effects of different limiting processes and front-tracking techniques.

Locally Limited and Fully Conserved RKDG2 Shallow Water Solutions with Wetting and Drying

This work extends a well-balanced second-order Runge-Kutta discontinuous Galerkin (RKDG2) scheme to provide conservative simulations for shallow flows involving wetting and drying over irregular topographies with friction effects. For this purpose, a wetting and drying technique designed originally for a finite volume (FV) scheme is improved and implemented, which includes the discretization of friction source terms via a splitting implicit integration approach. Another focus of this work is to design a fully conserved RKDG2 scheme to provide conservative solutions for both mass and momentum through a local slope limiting process. Several steady and transient benchmark tests with/without friction effects are simulated to validate the new solver and demonstrate the effects of different slope limiting processes, i.e. globally and locally slope limiting processes.

Well-balanced RKDG2 solutions to the shallow water equations over irregular domains with wetting and drying

This paper presents a new one-dimensional (1D) second-order Runge–Kutta discontinuous Galerkin (RKDG2) scheme for shallow flow simulations involving wetting and drying over complex domain topography. The shallow water equations that adopt water level (instead of water depth) as a flow variable are solved by an RKDG2 scheme to give piecewise linear approximate solutions, which are locally defined by an average coefficient and a slope coefficient. A wetting and drying technique proposed originally for a finite volume MUSCL scheme is revised and implemented in the RKDG2 solver. Extra numerical enhancements are proposed to amend the local coefficients associated with water level and bed elevation in order to maintain the well-balanced property of the RKDG2 scheme for applications with wetting and drying. Friction source terms are included and evaluated using splitting implicit discretization, implemented with a physical stopping condition to ensure stability. Several steady and unsteady benchmark tests with/without friction effects are considered to demonstrate the performance of the present model.

Accurate modelling of unsteady flows in collapsible tubes

The context of this paper is the development of a general and efficient numerical haemodynamic tool to help clinicians and researchers in understanding of physiological flow phenomena. We propose an accurate one-dimensional Runge–Kutta discontinuous Galerkin (RK-DG) method coupled with lumped parameter models for the boundary conditions. The suggested model has already been successfully applied to haemodynamics in arteries and is now extended for the flow in collapsible tubes such as veins. The main difference with cardiovascular simulations is that the flow may become supercritical and elastic jumps may appear with the numerical consequence that scheme may not remain monotone if no limiting procedure is introduced. We show that our second-order RK-DG method equipped with an approximate Roe's Riemann solver and a slope-limiting procedure allows us to capture elastic jumps accurately. Moreover, this paper demonstrates that the complex physics associated with such flows is more accurately modelled than with traditional methods such as finite difference methods or finite volumes. We present various benchmark problems that show the flexibility and applicability of the numerical method. Our solutions are compared with analytical solutions when they are available and with solutions obtained using other numerical methods. Finally, to illustrate the clinical interest, we study the emptying process in a calf vein squeezed by contracting skeletal muscle in a normal and pathological subject. We compare our results with experimental simulations and discuss the sensitivity to parameters of our model.

Well‐balancing issues related to the RKDG2 scheme for the shallow water equations

AbstractDiscontinuous Galerkin (DG) finite element methods have salient features that are mainly highlighted by their locality, their easiness in balancing the flux and source term gradients and their component‐wise structure. In the light of this, this paper aims to provide insights into the well‐balancing property of a second‐order Runge–Kutta Discontinuous Galerkin (RKDG2) method. For this purpose, a Godunov‐type RKDG2 method is presented for solving the shallow water equations. The scheme is based on local DG linear approximations and does not entail any special treatment of the source terms in order to achieve well‐balanced numerical results. The performance of the present RKDG2 scheme in reproducing conserved solutions for both free surface and discharge over strongly irregular topography is demonstrated by applying to several hydraulic benchmarks. Meanwhile, the effects of different slope limiting procedures on the well‐balancing property are investigated and discussed. This work may provide useful guidelines for developing a well‐balanced RKDG2 numerical scheme for shallow water flow simulation. Copyright © 2009 John Wiley & Sons, Ltd.

Scalar advection schemes for ocean modelling on unstructured triangular grids

Several schemes for scalar advection on unstructured triangular grids are assessed for possible use in ocean modelling applications. Finite element, finite volume and finite volume–element approaches are evaluated. A series of tests, including a numerical order of convergence analysis, idealized rotating cone and cylinder experiments, and transport of a tracer through the Stommel Gyre representation of ocean basin-scale circulation, are carried out. Volume element Eulerian–Lagrangian and third-order Runge-Kutta discontinuous Galerkin schemes are recommended for use in tracer studies. Taylor–Galerkin and second-order Runge–Kutta discontinuous Galerkin are found to be robust and accurate second-order schemes. When positivity is required, a fluctuation redistribution scheme was found to be an easily implemented, accurate, and computationally efficient approach.