High Order ADER Schemes for Continuum Mechanics

Busto, Saray; Chiocchetti, Simone; Dumbser, Michael; Gaburro, Elena; Peshkov, Ilya

doi:10.3389/fphy.2020.00032

Cited by 68 publications

(99 citation statements)

References 183 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The computational setup can be done in a fully automatic manner, without the need to generate a body-fitted structured or unstructured mesh, which can become very time consuming for complex geometries. The work presented in this paper is a natural extension of the simple diffuse interface approach introduced in [30,31,59,129] and [14]. The presented numerical algorithm with the underlying diffuse interface model is a consequent application of Godunov's shock capturing ideas to the context of moving material interfaces.…”

Section: Resultsmentioning

confidence: 99%

“…Now, the system (41), which contains only volume integrals to be calculated inside Ω i jk and no surface integrals, can be solved via a simple discrete Picard iteration for each element Ω i jk , and there is no need of any communication with neighbor elements. We recall that this procedure has been introduced for the first time in [32] for unstructured meshes, it has been extended for example to moving meshes in [7] and to degenerate space time elements in [58]; finally, its convergence has been formally proved in [14]. 10…”

Section: High Order In Time Via An Element-local Space-time Discontinmentioning

confidence: 99%

See 1 more Smart Citation

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

et al. 2020

Self Cite

View full text Add to dashboard Cite

In this paper we propose a new diffuse interface model for the numerical simulation of inviscid compressible flows around fixed and moving solid bodies of arbitrary shape. The solids are assumed to be moving rigid bodies, without any elastic properties. The mathematical model is a simplified case of the seven-equation Baer-Nunziato model of compressible multi-phase flows. The resulting governing PDE system is a nonlinear system of hyperbolic conservation laws with non-conservative products. The geometry of the solid bodies is simply specified via a scalar field that represents the volume fraction of the fluid present in each control volume. This allows the discretization of arbitrarily complex geometries on simple uniform or adaptive Cartesian meshes. Inside the solid bodies, the fluid volume fraction is zero, while it is unitary inside the fluid phase. Due to the diffuse interface nature of the model, the volume fraction function can assume any value between zero and one in mixed cells that are occupied by both, fluid and solid.We also prove that at the material interface, i.e. where the volume fraction jumps from unity to zero, the normal component of the fluid velocity assumes the value of the normal component of the solid velocity. This result can be directly derived from the governing equations, either via Riemann invariants or from the generalized Rankine Hugoniot conditions according to the theory of Dal Maso, Le Floch and Murat [89], which justifies the use of a path-conservative approach for treating the non-conservative products.The governing partial differential equations of our new model are solved on simple uniform Cartesian grids via a high order path-conservative ADER discontinuous Galerkin (DG) finite element method with a posteriori sub-cell finite volume (FV) limiter. Since the numerical method is of the shock capturing type, the fluid-solid boundary is never explicitly tracked by the numerical method, neither via interface reconstruction, nor via mesh motion.The effectiveness of the proposed approach is tested on a set of different numerical test problems, including 1D Riemann problems as well as supersonic flows over fixed and moving rigid bodies.Key words: diffuse interface model, compressible flows over fixed and moving solids, immersed boundary method for compressible flows, arbitrary high-order discontinuous Galerkin schemes, a posteriori sub-cell finite volume limiter (MOOD), path-conservative schemes for hyperbolic PDE with non-conservative products,

show abstract

Section: Resultsmentioning

confidence: 99%

Section: High Order In Time Via An Element-local Space-time Discontinmentioning

confidence: 99%

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

et al. 2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…Furthermore, in the specific case N = 0, i.e. when the P N P M reduces to a FV scheme, the above polynomial reconstruction procedure must be made nonlinear; this can be easily done, for example, by adopting the WENO strategy specifically described in the context of P N P M type schemes (thus with the same notation adopted here) on Cartesian meshes in [25,51]. We recall that the nonlinearity introduced through ENO/WENO type procedures essentially avoids the spurious oscillations typical of high order linear schemes modeling discontin-…”

Section: High Order Spatial Reconstruction (Order M)mentioning

confidence: 99%

“…Our limiter is based on the MOOD approach [29,35,36], which has already been successfully applied in the framework of ADER finite volume schemes [21,22,85] and Discontinous Galerkin finite element schemes, see [25,47,52,75,97,124]. Specifically, the numerical solution is checked a posteriori for nonphysical values and spurious oscillations, and if it does not satisfy all admissibility detection criteria, given by both physical and numerical requirements, in a certain cell, that cell is marked as troubled.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

Gaburro

Dumbser

2021

J Sci Comput

Self Cite

View full text Add to dashboard Cite

In this work, we consider the general family of the so called ADER $$P_NP_M$$ P N P M schemes for the numerical solution of hyperbolic partial differential equations with arbitrary high order of accuracy in space and time. The family of one-step $$P_NP_M$$ P N P M schemes was introduced in Dumbser (J Comput Phys 227:8209–8253, 2008) and represents a unified framework for classical high order Finite Volume (FV) schemes ($$N=0$$ N = 0 ), the usual Discontinuous Galerkin (DG) methods ($$N=M$$ N = M ), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degree M is applied over piecewise polynomial data of degree N with $$M>N$$ M > N . In all cases with $$M \ge N > 0 $$ M ≥ N > 0 the $$P_NP_M$$ P N P M schemes are linear in the sense of Godunov (Math. USSR Sbornik 47:271–306, 1959), thus when considering phenomena characterized by discontinuities, spurious oscillations may appear and even destroy the simulation. Therefore, in this paper we present a new simple, robust and accurate a posteriori subcell finite volume limiting strategy that is valid for the entire class of $$P_NP_M$$ P N P M schemes. The subcell FV limiter is activated only where it is needed, i.e. in the neighborhood of shocks or other discontinuities, and is able to maintain the resolution of the underlying high order $$P_NP_M$$ P N P M schemes, due to the use of a rather fine subgrid of $$2N+1$$ 2 N + 1 subcells per space dimension. The paper contains a wide set of test cases for different hyperbolic PDE systems, solved on adaptive Cartesian meshes that show the capabilities of the proposed method both on smooth and discontinuous problems, as well as the broad range of its applicability. The tests range from compressible gasdynamics over classical MHD to relativistic magnetohydrodynamics.

show abstract

A high‐order numerical method for sediment transport problems simulation and its comparison with laboratory experiments

Capilla

Balaguer-Beser

Nácher-Rodríguez

et al. 2021

Comp and Math Methods

View full text Add to dashboard Cite

This article describes a high-order well-balanced central finite volume scheme for solving the coupled Exner−shallow water equations in one dimensional channels with rectangular section and variable width. Such numerical method may solve the proposed bedload sediment transport problem without the need to diagonalize the Jacobian matrix of flow. The numerical scheme uses a Runge-Kutta method with a fourth-order continuous natural extension for time discretization. The source term approximation is designed to verify the exact conservation property. Comparison of the numerical results for two accuracy tests have proved the stability and accuracy of the scheme. The results of the laboratory tests have also been used to calibrate different expressions of the solid transport discharge in the computer code. Two experimental tests have been carried out to study the erosive phenomenon and the consequent sediment transport: one test consisting of a triangular dune, and other caused by the effect of channel contraction.

show abstract

High Order ADER Schemes for Continuum Mechanics

Cited by 68 publications

References 183 publications

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

A high‐order numerical method for sediment transport problems simulation and its comparison with laboratory experiments

Contact Info

Product

Resources

About