Solution of Compressible Euler Flows Using Rational Runge-Kutta Time Stepping Scheme

1989 ◽  
pp. 309-330 ◽  
Author(s):  
Nobuyuki Satofuka ◽  
Koji Morinishi
Keyword(s):  
Time Stepping ◽  
Compressible Euler ◽  
Euler Flows ◽  
Runge Kutta

scholarly journals Multigrid and Runge-Kutta time stepping applied to the uniformly non-oscillatory scheme for conservation laws

1991 ◽  
Vol 25 (3) ◽  
pp. 243-263 ◽  
Author(s):  
J. W. van der Burg ◽  
J. G. M. Kuerten ◽  
P. J. Zandbergen
Keyword(s):  
Conservation Laws ◽  
Time Stepping ◽  
Runge Kutta

scholarly journals Linear Stability Analysis of Runge–Kutta-Based Partial Time-Splitting Schemes for the Euler Equations

2010 ◽  
Vol 138 (12) ◽  
pp. 4475-4496 ◽  
Author(s):  
Michael Baldauf
Keyword(s):  
Stability Analysis ◽  
Euler Equations ◽  
Simulation Models ◽  
Euler System ◽  
Splitting Schemes ◽  
Stability Properties ◽  
Compressible Euler ◽  
Numerical Stability Analysis ◽  
Time Splitting ◽  
Runge Kutta

Abstract For atmospheric simulation models with resolutions from about 10 km to the subkilometer cloud-resolving scale, the complete nonhydrostatic compressible Euler equations are often used. An important integration technique for them is the time-splitting (or split explicit) method. This article presents a comprehensive numerical stability analysis of Runge–Kutta (RK)-based partial time-splitting schemes. To this purpose a linearized two-dimensional (2D) compressible Euler system containing advection (as the slow process), sound, and gravity wave terms (as fast processes) is considered. These processes are the most important ones in limiting stability. First, the detailed stability properties are discussed with regard to several off-centering weights for each fast process described by horizontally explicit, vertically implicit schemes. Then the stability properties of the temporally and spatially discretized three-stage RK scheme for the complete 2D Euler equations and their stabilization (e.g., by divergence damping) are discussed. The main goal is to find optimal values for all of the occurring numerical parameters to guarantee stability in operational model applications. Furthermore, formal orders of temporal truncation errors for the time-splitting schemes are calculated. With the same methodology, two alternatives to the three-stage RK method, a so-called RK3-TVD method, and a new four-stage, second-order RK method are inspected.

Runge--Kutta-Based Explicit Local Time-Stepping Methods for Wave Propagation

2015 ◽  
Vol 37 (2) ◽  
pp. A747-A775 ◽  
Author(s):  
Marcus J. Grote ◽  
Michaela Mehlin ◽  
Teodora Mitkova
Keyword(s):  
Wave Propagation ◽  
Local Time ◽  
Time Stepping ◽  
Local Time Stepping ◽  
Runge Kutta

scholarly journals A Runge Kutta Discontinuous Galerkin Method for Lagrangian Compressible Euler Equations in Two-Dimensions

2014 ◽  
Vol 15 (4) ◽  
pp. 1184-1206 ◽  
Author(s):  
Zhenzhen Li ◽  
Xijun Yu ◽  
Jiang Zhu ◽  
Zupeng Jia
Keyword(s):  
Euler Equations ◽  
Discontinuous Galerkin ◽  
Gas Dynamics ◽  
Two Dimensions ◽  
Oscillatory Property ◽  
Order Accuracy ◽  
Numerical Tests ◽  
Compressible Euler ◽  
Free Parameters ◽  
Runge Kutta

AbstractThis paper presents a new Lagrangian type scheme for solving the Euler equations of compressible gas dynamics. In this new scheme the system of equations is discretized by Runge-Kutta Discontinuous Galerkin (RKDG) method, and the mesh moves with the fluid flow. The scheme is conservative for the mass, momentum and total energy and maintains second-order accuracy. The scheme avoids solving the geometrical part and has free parameters. Results of some numerical tests are presented to demonstrate the accuracy and the non-oscillatory property of the scheme.

scholarly journals Generalized Split-Explicit Runge–Kutta Methods for the Compressible Euler Equations

2014 ◽  
Vol 142 (5) ◽  
pp. 2067-2081 ◽  
Author(s):  
Oswald Knoth ◽  
Joerg Wensch
Keyword(s):  
Euler Equations ◽  
Time Integration ◽  
Order Conditions ◽  
Compressible Euler Equations ◽  
Step Size ◽  
Integration Interval ◽  
New Methods ◽  
Runge Kutta Method ◽  
Compressible Euler ◽  
Runge Kutta

Abstract The compressible Euler equations exhibit wave phenomena on different scales. A suitable spatial discretization results in partitioned ordinary differential equations where fast and slow modes are present. Generalized split-explicit methods for the time integration of these problems are presented. The methods combine explicit Runge–Kutta methods for the slow modes and with a free choice integrator for the fast modes. Order conditions for these methods are discussed. Construction principles to develop methods with enlarged stability area are presented. Among the generalized class several new methods are developed and compared to the well-established three-stage low-storage Runge–Kutta method (RK3). The new methods allow a 4 times larger macro step size. They require a smaller integration interval for the fast modes. Further, these methods satisfy the order conditions for order three even for nonlinear equations. Numerical tests on more complex problems than the model equation confirm the enhanced stability properties of these methods.

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