scholarly journals Analysis of chosen numerical methods for the application in high order interior ballistics simulations

2021 ◽  
Vol 2090 (1) ◽  
pp. 012112
Author(s):  
M Krol
Keyword(s):  
High Order ◽  
Test Problems ◽  
Numerical Accuracy ◽  
Order Of Accuracy ◽  
Computing Power ◽  
Interior Ballistics ◽  
Depth Analysis ◽  
Catastrophic Failures ◽  
Volume Method ◽  
The Mathematical Model

Abstract Considering constant development of the interior ballistics, along with new gun and ammunition designs, the necessity of in-depth analysis of the shot event is continuously increasing. Numerical simulations of interior ballistics problems are useful for optimising new designs or explaining complex issues, regarding performance instabilities and catastrophic failures. With the rise of the computing power, there is a significant urge to drive the numerical errors towards machine zero. This goal demands using methods of high order of accuracy in both space and time. Current methods allow to achieve an arbitrary order of numerical accuracy, thus allowing to shift the focus towards sophistication of the mathematical model of the studied phenomenon. Therefore, in this work, some numerical schemes, in context of finite volume method, are reviewed and studied using well established test problems. The results of the presented analysis are meant to become the basis for future development of a high order numerical scheme for simulation of interior ballistics problems.

An Essentially Non-Oscillatory Spectral Deferred Correction Method for Conservation Laws

2016 ◽  
Vol 13 (05) ◽  
pp. 1650027 ◽  
Author(s):  
Samet Y. Kadioglu ◽  
Veli Colak
Keyword(s):  
Conservation Laws ◽  
Fourth Order ◽  
Time Integration ◽  
High Order ◽  
Test Problems ◽  
Order Of Accuracy ◽  
Volume Method ◽  
Spectral Deferred Correction ◽  
Spectral Deferred Corrections ◽  
Eno Scheme

We present a computational method based on the Spectral Deferred Corrections (SDC) time integration technique and the Essentially Non-Oscillatory (ENO) finite volume method for the conservation laws (one-dimensional Euler equations). The SDC technique is used to advance the solutions in time with high-order of accuracy. The ENO method is used to define high-order cell edge quantities that are then used to evaluate numerical fluxes. The coupling of the SDC method with a high-order finite volume method (Piece-wise Parabolic Method (PPM)) for solving the conservation laws is first carried out by Layton et al. in [Layton, A. T. and Minion, M. L. [2004] “Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics,” J. Comput. Phys. 194(2), 697–714]. Issues about this approach have been addressed and some improvements have been added to it in [Kadioglu et al. [2012] “A gas dynamics method based on the spectral deferred corrections (SDC) time integration technique and the piecewise parabolic method (PPM),” Am. J. Comput. Math. 1–4, 303–317]. Here, we investigate the implications when the PPM method is replaced with the well-known ENO method. We note that the SDC-PPM method is fourth-order accurate in time and space. Therefore, we kept the order of accuracy of the ENO procedure as fourth-order in order to be able to make a consistent comparison between the two approaches (SDC-ENO versus SDC-PPM methods). We have tested the new SDC-ENO technique by solving several test problems involving moderate to strong shock waves and smooth/complex flow structures. Our numerical results show that we have numerically achieved the formally fourth-order convergence of the new method for smooth problems. Our numerical results also indicate that the newly proposed technique performs very well providing highly resolved shock discontinuities and fairly good contact solutions. More importantly, the discontinuities in the flow test problems are captured with essentially no-oscillations. We have numerically compared the fourth-order SDC-ENO scheme to the fourth-order SDC-PPM method for the same test problems. The results are similar for most of the test problems except in some cases the SDC-PPM method suffers from minor oscillations compared to SDC-ENO scheme being completely oscillation free.

An Essentially Nonoscillatory Spectral Deferred Correction Method for Hyperbolic Problems

2016 ◽  
Vol 13 (03) ◽  
pp. 1650017 ◽  
Author(s):  
Samet Y. Kadioglu
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Fourth Order ◽  
High Order ◽  
Computational Method ◽  
Hyperbolic Problems ◽  
Discontinuous Solutions ◽  
Order Of Accuracy ◽  
Volume Method ◽  
Essentially Nonoscillatory

We present a computational method based on the spectral deferred corrections (SDC) time integration technique and the essentially nonoscillatory (ENO) finite volume method for hyperbolic problems. The SDC technique is used to advance the solutions in time with high-order of accuracy. The ENO method is used to define high-order cell edge quantities that are then used to evaluate numerical fluxes. The coupling of the SDC method with a high-order finite volume method (piece-wise parabolic method (PPM)) is first carried out by Layton et al. [J. Comput. Phys. 194(2) (2004) 697]. Issues about this approach have been addressed and some improvements have been added to it in Kadioglu et al. [J. Comput. Math. 1(4) (2012) 303]. Here, we investigate the implications when the PPM method is replaced with the well-known ENO method. We note that the SDC-PPM method is fourth-order accurate in time and space. Therefore, we kept the order of accuracy of the ENO procedure as fourth-order in order to be able to make a consistent comparison between the two approaches (SDC-ENO versus SDC-PPM). We have tested the new SDC-ENO technique by solving smooth and nonsmooth hyperbolic problems. Our numerical results indicate that the fourth-order of accuracy in both space and time has been achieved for smooth problems. On the other hand, the new method performs very well when it is applied to nonlinear problems that involve discontinuous solutions. In other words, we have obtained highly resolved discontinuous solutions with essentially no-oscillations at or around the discontinuities.

scholarly journals Grid Average and Nodal Value in High Order Scheme

2018 ◽  
Vol 35 (3) ◽  
pp. 335-342
Author(s):  
Z. Liu ◽  
Q. Cai
Keyword(s):  
Truncation Error ◽  
Absolute Error ◽  
High Order ◽  
Fine Grid ◽  
Order Of Accuracy ◽  
High Order Scheme ◽  
High Order Schemes ◽  
Order Scheme ◽  
The Absolute ◽  
Volume Method

ABSTRACTThe concepts of nodal value and grid average in cell centered finite volume method (FVM) are clarified in this work, strict distinction between the two concepts in constructing numerical schemes is made, and common fault in misidentifying the two concepts is pointed out. The expansion based on grid average, similar to Taylor’s expansion, is deduced to construct correct scheme in terms of grid average and to obtain modified partial differential equation (MPDE) which determines the order of accuracy of numerical scheme theoretically. Correct high order scheme, taking QUICK (Quadratic Upstream Interpolation for Convective Kinematics) scheme as an example, is constructed in different approaches. Furthermore, the property of interpolation coefficients is analyzed. We also pointed out that for high order schemes, round-off error dominates the absolute error in fine grid and truncation error dominates the absolute error in coarse grid.

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.

Numerical methods with a high order of accuracy applied in the quantum system

1996 ◽  
Vol 104 (6) ◽  
pp. 2275-2286 ◽  
Author(s):  
Wusheng Zhu ◽  
Xinsheng Zhao ◽  
Youqi Tang
Keyword(s):  
Numerical Methods ◽  
Quantum System ◽  
High Order ◽  
Order Of Accuracy ◽  
High Order Of Accuracy

scholarly journals Temporal Evolution of Cooling by Natural Convection in an Enclosed Magma Chamber

Processes ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 108
Author(s):  
Carlos Enrique Zambra ◽  
Luciano Gonzalez-Olivares ◽  
Johan González ◽  
Benjamin Clausen
Keyword(s):  
Natural Convection ◽  
Magma Chamber ◽  
Temporal Evolution ◽  
Basaltic Magma ◽  
Boussinesq Equations ◽  
Rhyolitic Magma ◽  
The Times ◽  
Volume Method ◽  
The Mathematical Model ◽  
Liquid Magma

This research numerically studies the transient cooling of partially liquid magma by natural convection in an enclosed magma chamber. The mathematical model is based on the conservation laws for momentum, energy and mass for a non-Newtonian and incompressible fluid that may be modeled by the power law and the Oberbeck–Boussinesq equations (for basaltic magma) and solved with the finite volume method (FVM). The results of the programmed algorithm are compared with those in the literature for a non-Newtonian fluid with high apparent viscosity (10–200 Pa s) and Prandtl (Pr = 4 × 104) and Rayleigh (Ra = 1 × 106) numbers yielding a low relative error of 0.11. The times for cooling the center of the chamber from 1498 to 1448 K are 40 ky (kilo years), 37 and 28 ky for rectangular, hybrid and quasi-elliptical shapes, respectively. Results show that for the cases studied, natural convection moved the magma but had no influence on the isotherms; therefore the main mechanism of cooling is conduction. When a basaltic magma intrudes a chamber with rhyolitic magma in our model, natural convection is not sufficient to effectively mix the two magmas to produce an intermediate SiO2 composition.

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