Essentially non-oscillatory and weighted essentially non-oscillatory schemes

Shu, Chi-Wang

doi:10.1017/s0962492920000057

Cited by 118 publications

(46 citation statements)

References 289 publications

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“…[61] in which a submerged flexible structure is interacting with the surrounding compressible fluid, more specifically, air. In this case, fully-fledged energy conservation form of governing equations in aerodynamics based on conformal mapping and compact schemes have been coupled with non-oscillatory time incremental schemes [60,61]. The specific details are imbedded in the proprietary code in Wright-Patterson Air Force Base.…”

Section: Results and Verificationsmentioning

confidence: 99%

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A Revisit of Implicit Monolithic Algorithms for Compressible Solids Immersed Inside a Compressible Liquid

Wang

2021

Fluids

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With the development of mature Computational Fluid Dynamics (CFD) tools for fluids (air and liquid) and Finite Element Methods (FEM) for solids and structures, many approaches have been proposed to tackle the so-called Fluid–Structure Interaction or Fluid–Solid Interaction (FSI) problems. Traditional partitioned iterations are often used to link available FEM codes with CFD codes in the study of FSI systems. Although these procedures are convenient, fluid mesh adjustments according to the motion and finite deformation of immersed solids or structures can be challenging or even prohibitive. Moreover, complex dynamic behaviors of coupled FSI systems are often lost in these iterative processes. In this paper, the author would like to review the so-called monolithic approaches for the solution of coupled FSI systems as a whole in the context of the immersed boundary method. In particular, the focus is on the implicit monolithic algorithm for compressible solids immersed inside a compressible liquid. Notice here the main focus of this paper is on liquid or more precisely liquid phase of water as working fluid. Using the word liquid, the author would like to emphasize the consideration of the compressibility of the fluid and the assumption of constant density and temperature. It is a common practice to assume that the pressure variations are not strong enough to alter the liquid density in any significant fashion for acoustic fluid–solid interactions problems. Although the algorithm presented in this paper is not directly applicable to aerodynamics in which the density change is significant along with its relationship with the pressure and the temperature, the author did revisit his earlier work on merging immersed boundary method concepts with a fully-fledged compressible aerodynamic code based on high-order compact scheme and energy conservative form of governing equations. In the proposed algorithm, on top of a uniform background (ghost) mesh, a fully implicit immersed method is implemented with mixed finite element methods for compressible liquid as well as immersed compressible solids with a matrix-free Newton–Krylov iterative solution scheme. In this monolithic approach, with the simple modulo function, the immersed solid or structure points can be easily located and thus the displacement projections and force distributions stipulated in the immersed boundary method can be effectively and efficiently implemented. This feature coupled with the key concept of the immersed boundary method helps to avoid topologically challenging mesh adjustments and to incorporate parallel processing commands such as Message Passing Interface (MPI) and further vectorization of the numerical operation. Once these high-performance procedures are implemented coupled with the monolithic implicit matrix-free Newton–Krylov iterative scheme with immersed methods, effective and efficient reduced order modeling techniques can then be employed to explore phase and parametric spaces. The in-house developed programs are at the moment two-dimensional. Furthermore, based on the same approach implemented in one-dimensional test example with one continuum immersed in another continuum, such monolithic implicit matrix-free Newton–Krylov iterative approach can be extended for the study of composites with deformable aggregates and matrix.

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Section: Results and Verificationsmentioning

confidence: 99%

“…The change of the water density is very minimum and should be ignored, so is the density of the immersed solid. Notice that when we deal with the air as the working fluid, fully-fledged energy conservation form of governing equations must be used along with non-oscillatory time incremental schemes [60,61].…”

Section: Mixed Finite Element Formulationsmentioning

confidence: 99%

A Revisit of Implicit Monolithic Algorithms for Compressible Solids Immersed Inside a Compressible Liquid

Wang

2021

Fluids

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“…∑ i=−2,…,2 i k+1 y j L i,j (y(t)) , k = 0, 1, 2, and denote by w ± i,m the weights that correspond to the stencils {x m−1 , x m , x m+1 } , m = j − 1, j, j + 1 . From ( 22), we calculate which gives by using ( 26), (27) for different stencils and Lemma 3.1 for all terms…”

Section: Proof Let Us Define D K ∶=mentioning

confidence: 99%

“…Although these methods are formally firstorder accurate when a shock is present, they still have uniform high-order accuracy right up to the shock location. WENO schemes are extensions of the ENO procedure, i.e., they perform essentially non-oscillatory, but overcome shortcomings of the ENO approximation, see [27] for a detailed discussion. By employing a global flux-splitting, the numerical flux function becomes classically differentiable and therefore allows to develop discrete adjoint WENO methods of higher order.…”

Section: Introductionmentioning

confidence: 99%

A third-order weighted essentially non-oscillatory scheme in optimal control problems governed by nonlinear hyperbolic conservation laws

Frenzel

Lang

2021

Comput Optim Appl

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The weighted essentially non-oscillatory (WENO) methods are popular and effective spatial discretization methods for nonlinear hyperbolic partial differential equations. Although these methods are formally first-order accurate when a shock is present, they still have uniform high-order accuracy right up to the shock location. In this paper, we propose a novel third-order numerical method for solving optimal control problems subject to scalar nonlinear hyperbolic conservation laws. It is based on the first-disretize-then-optimize approach and combines a discrete adjoint WENO scheme of third order with the classical strong stability preserving three-stage third-order Runge–Kutta method SSPRK3. We analyze its approximation properties and apply it to optimal control problems of tracking-type with non-smooth target states. Comparisons to common first-order methods such as the Lax–Friedrichs and Engquist–Osher method show its great potential to achieve a higher accuracy along with good resolution around discontinuities.

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“…Their design encounters the main difficulties in controlling spurious oscillations near discontinuities and near the domain boundaries. The first problem is well tackled by reconstructions of the weighted essentially non-oscillatory ( ) class introduced in (see the reviews [36][37][38]) or by the central weighted essentially non-oscillatory [35] setting (…”

Section: Introductionmentioning

confidence: 99%

One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

Semplice

Travaglia

Puppo

2021

Commun. Appl. Math. Comput.

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We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in the boundary cells. In Naumann et al. (Appl. Math. Comput. 325: 252–270. 10.1016/j.amc.2017.12.041, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests, we compare the novel reconstruction with the standard approach using ghost cells.

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Essentially non-oscillatory and weighted essentially non-oscillatory schemes

Cited by 118 publications

References 289 publications

A Revisit of Implicit Monolithic Algorithms for Compressible Solids Immersed Inside a Compressible Liquid

A Revisit of Implicit Monolithic Algorithms for Compressible Solids Immersed Inside a Compressible Liquid

A third-order weighted essentially non-oscillatory scheme in optimal control problems governed by nonlinear hyperbolic conservation laws

One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

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