Error Estimates for the Runge–Kutta Discontinuous Galerkin Method for the Transport Equation with Discontinuous Initial Data

2008 ◽  
Vol 46 (3) ◽  
pp. 1364-1398 ◽  
Author(s):  
Bernardo Cockburn ◽  
Johnny Guzmán
Keyword(s):  
Initial Data ◽  
Galerkin Method ◽  
Transport Equation ◽  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Discontinuous Initial Data ◽  
Runge Kutta

Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws

2019 ◽  
Vol 53 (1) ◽  
pp. 105-144 ◽  
Author(s):  
Lingling Zhou ◽  
Yinhua Xia ◽  
Chi-Wang Shu
Keyword(s):  
Conservation Laws ◽  
Galerkin Method ◽  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Second Order ◽  
Arbitrary Lagrangian Eulerian ◽  
Third Order ◽  
Fully Discrete ◽  
Runge Kutta

In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh2 and the second order TVD-RK scheme needs $ \tau \le \rho {h}^{\frac{4}{3}}$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h.

Runge-Kutta Discontinuous Galerkin Method with Front Tracking Method for Solving the Compressible Two-Medium Flow on Unstructured Meshes

2016 ◽  
Vol 9 (1) ◽  
pp. 73-91 ◽  
Author(s):  
Haitian Lu ◽  
Jun Zhu ◽  
Chunwu Wang ◽  
Ning Zhao
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Front Tracking ◽  
Unstructured Meshes ◽  
Interfacial Region ◽  
Tracking Method ◽  
Front Tracking Method ◽  
Medium Flow ◽  
Runge Kutta

AbstractIn this paper, we extend using the Runge-Kutta discontinuous Galerkin method together with the front tracking method to simulate the compressible two-medium flow on unstructured meshes. A Riemann problem is constructed in the normal direction in the material interfacial region, with the goal of obtaining a compact, robust and efficient procedure to track the explicit sharp interface precisely. Extensive numerical tests including the gas-gas and gas-liquid flows are provided to show the proposed methodologies possess the capability of enhancing the resolutions nearby the discontinuities inside of the single medium flow and the interfacial vicinities of the two-medium flow in many occasions.

scholarly journals Discontinuous Galerkin method for solving incompressible two-phase shallow granular flow model

2019 ◽  
Vol 11 (9) ◽  
pp. 168781401987490
Author(s):  
Muhammad Rehan Saleem ◽  
Ubaid Ahmed Nisar ◽  
Shamsul Qamar
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Numerical Scheme ◽  
Flow Model ◽  
Numerical Study ◽  
Momentum Exchange ◽  
Two Phase ◽  
Runge Kutta ◽  
Model Equations

This article deals with the numerical study of two-phase shallow flow model describing the mixture of fluid and solid granular particles. The model under investigation consists of coupled mass and momentum equations for solid granular material and fluid particles through non-conservative momentum exchange terms. The non-conservativity of model equations poses major challenges for any numerical scheme, such as well balancing, positivity preservation, accurate approximation of non-conservative terms, and achievement of steady-state conditions. Thus, in order to approximate the present model an accurate, well-balanced, robust, and efficient numerical scheme is required. For this purpose, in this article, Runge–Kutta discontinuous Galerkin method is applied successfully for the first time to solve the model equations. Several test problems are also carried out to check the performance and accuracy of our proposed numerical method. To compare the results, the same model is solved by staggered central Nessyahu–Tadmor scheme. A good comparison is found between two schemes, but the results obtained by Runge–Kutta discontinuous Galerkin scheme are found superior over the central Nessyahu–Tadmor scheme.

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