Finite volume and discontinous Galerkin schemes for stiff source term problems

2012 ◽  
pp. 309-309
Keyword(s):  
Finite Volume ◽  
Source Term ◽  
Discontinous Galerkin

scholarly journals A finite volume method for two-moment cosmic-ray hydrodynamics on a moving mesh

2021 ◽  
Author(s):  
T Thomas ◽  
C Pfrommer ◽  
R Pakmor
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Source Term ◽  
Cosmic Ray ◽  
Moving Mesh ◽  
Test Problems ◽  
Integration Step ◽  
Custom Made ◽  
Anisotropic Transport ◽  
Volume Method

Abstract We present a new numerical algorithm to solve the recently derived equations of two-moment cosmic ray hydrodynamics (CRHD). The algorithm is implemented as a module in the moving mesh Arepo code. Therein, the anisotropic transport of cosmic rays (CRs) along magnetic field lines is discretised using a path-conservative finite volume method on the unstructured time-dependent Voronoi mesh of Arepo. The interaction of CRs and gyroresonant Alfvén waves is described by short-timescale source terms in the CRHD equations. We employ a custom-made semi-implicit adaptive time stepping source term integrator to accurately integrate this interaction on the small light-crossing time of the anisotropic transport step. Both the transport and the source term integration step are separated from the evolution of the magneto-hydrodynamical equations using an operator split approach. The new algorithm is tested with a variety of test problems, including shock tubes, a perpendicular magnetised discontinuity, the hydrodynamic response to a CR overpressure, CR acceleration of a warm cloud, and a CR blast wave, which demonstrate that the coupling between CR and magneto-hydrodynamics is robust and accurate. We demonstrate the numerical convergence of the presented scheme using new linear and non-linear analytic solutions.

scholarly journals Conservative finite-volume forms of the Saint-Venant equations for hydrology and urban drainage

2019 ◽  
Vol 23 (3) ◽  
pp. 1281-1304 ◽  
Author(s):  
Ben R. Hodges
Keyword(s):  
Free Surface ◽  
Finite Volume ◽  
Source Term ◽  
Weighting Function ◽  
Bottom Topography ◽  
Urban Drainage ◽  
Venant Equation ◽  
Saint Venant Equations ◽  
Volume Forms ◽  
Conservative Flux

Abstract. New integral, finite-volume forms of the Saint-Venant equations for one-dimensional (1-D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. This approach prevents irregular channel topography from creating an inherently non-smooth source term for momentum. The derivation introduces an analytical approximation of the free surface across a finite-volume element (e.g., linear, parabolic) with a weighting function for quadrature with bottom topography. This new free-surface/topography approach provides a single term that approximates the integrated piezometric pressure over a control volume that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting conservative finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, and water surface elevations – without using the bottom slope (S0). The new Saint-Venant equation form is (1) inherently conservative, as compared to non-conservative finite-difference forms, and (2) inherently well-balanced for irregular topography, as compared to conservative finite-volume forms using the Cunge–Liggett approach that rely on two integrations of topography. It is likely that this new equation form will be more tractable for large-scale simulations of river networks and urban drainage systems with highly variable topography as it ensures the inhomogeneous source term of the momentum conservation equation is Lipschitz smooth as long as the solution variables are smooth.

A numerical approach for simulating two-dimensional dense-snow avalanches in global coordinate systems

2021 ◽  
Author(s):  
Daniel Zugliani ◽  
Giorgio Rosatti ◽  
Stefania Sansone
Keyword(s):  
Coordinate System ◽  
Finite Volume ◽  
Reference System ◽  
Source Term ◽  
Numerical Approach ◽  
The Other ◽  
Snow Avalanche ◽  
Two Dimensional ◽  
Snow Avalanches ◽  
Avalanche Models

<p>Snow avalanche models are commonly based on a continuum fluid scheme, on the assumption of shallow flow in the direction normal to the bed, on a depth-averaged description of the flow quantities and on different assumptions concerning the velocity profile, the friction law, and the pressure in the flow direction (see Bartelt et al, 1999, Barbolini et al., 2000, for an overview). The coordinate reference system is commonly local, i.e., for each point of the domain, one axis is normal to the bed while the other two axes lay in a tangent plane. When the bed is vertical and the flow is not aligned with the steepest direction (e.g., in case of a side wall), the flow depth is no longer defined considering the normal direction and the model based on the local coordinate system is no longer valid. In near-vertical conditions, numerical problems can be expected.</p><p>Another critical point, for numerical models based on finite volume schemes and Godunov fluxes, is the accurate treatment of the source term in case of no-motion conditions (persistence, starting and stopping of the flow) due to the presence of velocity-independent, Coulomb-type terms in the bed shear stress. </p><p>In this work, we provide a numerical approach for a Voellmy-fluid based model, able to overcome the limits depicted above, to accurately simulate analytical solutions and to give reliable solutions in other cases (Zugliani & Rosatti, 2021). Firstly, differently from the other literature models, the chosen coordinate reference system is global (an axis opposite the gravity vector and the other two orthogonal axes lay in the horizontal plane) and therefore, the relevant mass and momentum equations have been derived accordingly. Secondly, these equations have been discretized by using a finite volume method on a Cartesian square grid where the Godunov fluxes has been evaluated by mean of a modified DOT scheme (Zugliani & Rosatti, 2016) while source terms in conditions of motion have been discretized by using an implicit operator-splitting technique. Finally, a specific algorithm has been derived to deal with the source term to determine the no-motion conditions.  Several test cases assess the capabilities of the proposed approach.</p><p> </p><p><strong>References:</strong></p><p>Barbolini, M., Gruber, U., Keylock, C.J., Naaim, M., Savi, F. (2000), <em>Application of statistical and hydraulic-continuum dense-snow avalanche models to five real European sites.</em> Cold Regions Science and Tech. 31, 133–149.</p><p>Bartelt, P., Salm, B., Gruber, U. (1999), <em>Calculating dense-snow avalanche runout using a voellmy-fluid model with active/passive longitudinal straining.</em> Journal of Glaciology 45, 242-254.</p><p>Zugliani D., Rosatti G. (2021), <em>Accurate modeling of two-dimensional dense snow avalanches in global coordinate system: the TRENT2D<sup>❄</sup> approach. </em>Paper under review.</p><p>Zugliani, D., Rosatti, G. (2016), <em>A new Osher Riemann solver for shallow water flow over fixed or mobile bed</em>, Proceedings of the 4th European Congress of the IAHR, pp. 707–713.</p>

scholarly journals Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

2011 ◽  
Vol 9 (2) ◽  
pp. 324-362 ◽  
Author(s):  
Franz Georg Fuchs ◽  
Andrew D. McMurry ◽  
Siddhartha Mishra ◽  
Nils Henrik Risebro ◽  
Knut Waagan
Keyword(s):  
Finite Volume ◽  
Source Term ◽  
Second Order ◽  
High Order ◽  
Mhd Equations ◽  
Positive Pressure ◽  
Finite Volume Schemes ◽  
Riemann Solvers ◽  
Ideal Mhd ◽  
Approximate Riemann Solvers

AbstractWe design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three- and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semi-conservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO- and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first- and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshes.

scholarly journals Effective Source Term Discretizations for Higher Accuracy Finite Volume Discretization of Parabolic Equations

2021 ◽  
Vol 9 (8) ◽  
pp. 366-392
Author(s):  
Yaw Kyei
Keyword(s):  
Finite Volume ◽  
Parabolic Equations ◽  
Source Term ◽  
Space Time ◽  
Local Source ◽  
Source Terms ◽  
Equation Error ◽  
Diffusive Fluxes ◽  
The Right ◽  
Volume Method

A finite volume method is applied to develop space-time discretizations for parabolic equations based on an equation error method.A space-time expansion of the local equation error based on flux integral formulation of the equation is first designed using a desiredframework of neighboring quadrature points for the solution and local source terms. The quadrature weights are then determined through aminimization process for the error which constitutes all local compact fluxes about each centroid within the computational domain.In utilizing a local source term distribution to account for diffusive fluxes, the right minimizing quadrature weights and collocationpoints including subgrid points for the source terms may be determined and optimized for higher accuracies as well as robust higher-ordercomputational convergence. The resulting local residuals form a more complete description of the truncation errors which are then utilizedto assess the computational performances of the resulting schemes. The effectiveness of the discretization method is demonstrated by theresults and analysis of the schemes.

A High Order Well-Balanced Finite Volume WENO Scheme for a Blood Flow Model in Arteries

2017 ◽  
Vol 7 (4) ◽  
pp. 852-866
Author(s):  
Zhonghua Yao ◽  
Gang Li ◽  
Jinmei Gao
Keyword(s):  
Blood Flow ◽  
Steady State ◽  
Finite Volume ◽  
Flow Model ◽  
Source Term ◽  
High Order ◽  
Weno Scheme ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
Steady State Solutions

AbstractThe numerical simulations for the blood flow in arteries by high order accurate schemes have a wide range of applications in medical engineering. The blood flow model admits the steady state solutions, in which the flux gradient is non-zero and is exactly balanced by the source term. In this paper, we present a high order finite volume weighted essentially non-oscillatory (WENO) scheme, which preserves the steady state solutions and maintains genuine high order accuracy for general solutions. The well-balanced property is obtained by a novel source term reformulation and discretisation, combined with well-balanced numerical fluxes. Extensive numerical experiments are carried out to verify well-balanced property, high order accuracy, as well as good resolution for smooth and discontinuous solutions.

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