Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes

1981 ◽  
Author(s):  
A. JAMESON ◽  
WOLFGANG SCHMIDT ◽  
ELI TURKEL
Keyword(s):  
Numerical Solution ◽  
Euler Equations ◽  
Finite Volume ◽  
Finite Volume Methods ◽  
Time Stepping ◽  
Runge Kutta

Advances in the Numerical Integration of the Three-Dimensional Euler Equations in Vibrating Cascades

1993 ◽  
Vol 115 (4) ◽  
pp. 781-790 ◽  
Author(s):  
G. A. Gerolymos
Keyword(s):  
Boundary Conditions ◽  
Numerical Integration ◽  
Euler Equations ◽  
Finite Volume ◽  
Three Dimensional ◽  
Computational Results ◽  
Grid Refinement ◽  
Mobile Grid ◽  
Transonic Compressor ◽  
Runge Kutta

In the present work an algorithm for the numerical integration of the three-dimensional unsteady Euler equations in vibrating transonic compressor cascades is described. The equations are discretized in finite-volume formulation in a mobile grid using isoparametric brick elements. They are integrated in time using Runge-Kutta schemes. A thorough discussion of the boundary conditions used and of their influence on results is undertaken. The influence of grid refinement on computational results is examined. Unsteady convergence of results is discussed.

Reprint of: Finite volume methods for multi-component Euler equations with source terms

2018 ◽  
Vol 169 ◽  
pp. 40-61
Author(s):  
Alfredo Bermúdez ◽  
Xián López ◽  
M. Elena Vázquez-Cendón
Keyword(s):  
Euler Equations ◽  
Finite Volume ◽  
Finite Volume Methods ◽  
Source Terms

scholarly journals Finite volume methods for multi-component Euler equations with source terms

2017 ◽  
Vol 156 ◽  
pp. 113-134 ◽  
Author(s):  
Alfredo Bermúdez ◽  
Xián López ◽  
M. Elena Vázquez-Cendón
Keyword(s):  
Euler Equations ◽  
Finite Volume ◽  
Finite Volume Methods ◽  
Source Terms

scholarly journals A Forward Time Stepping Heat Conduction Model for Spot Melt Additive Manufacturing

2019 ◽  
Vol 141 (11) ◽  
Author(s):  
B. Stump ◽  
A. Plotkowski
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Analytic Solution ◽  
Dimensional Space ◽  
Mathematical Formulation ◽  
Finite Volume Methods ◽  
Analytic Solutions ◽  
Conduction Model ◽  
Time Stepping ◽  
Speed And Accuracy

Abstract Solidification dynamics are crucial for determining microstructure development in additively manufactured parts. Multiphysics models based on finite element or finite volume methods may help gain insight for complicated phenomena such as fluid flow, keyholing, and porosity but are too computationally expensive to use for simulating actual builds. Recent analytic and semi-analytic solutions for moving heat sources in a semi-infinite three-dimensional space provide a way to accurately estimate the solidification conditions for entire builds. The downside to these methods is that, unlike finite element or finite volume methods, they cannot use the temperature distribution of the previous timesteps to march the solution forward in time. This paper provides the mathematical formulation and implementation of a forward time stepping (FTS) approach to an existing semi-analytic solution. The speed and accuracy of the two methods are then compared for various scan patterns. The result is that, for spot melts, the forward time-stepping model provides improvements in both speed and accuracy. This is especially true for longer simulations, where the simulation can be orders of magnitude faster. The longest simulation analyzed in this paper was roughly 30× faster when using the forward time-stepping model versus the straightforward implementation of the semi-analytic solution.

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