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We propose a new class of fourth- and sixth-order schemes in time for parabolic and hyperbolic equations. The method follows the compact scheme methodology by elaborating implicit relations between the approximations of the function and its derivatives. We produce a series of A-stable methods with low dispersion and high accuracy. Several benchmarks for linear and non-linear Ordinary Differential Equations demonstrate the effectiveness of the method. Then a second set of numerical benchmarks for Partial Differential Equations such as convection-diffusion, Schrödinger equation, wave equation, Bürgers, and Euler system give the numerical evidences of the superior advantage of the method with respect to the traditional Runge-Kutta or multistep methods. [ABSTRACT FROM AUTHOR]

Stability condition is a more restrictive constraint that leads to unnecessary small-time steps with respect to the accuracy and results in computational time wastage. We propose a node by node adaptive time scheme to relax the stability constraint enabling a larger global time step for all the nodes. A nonlinear procedure for optimising both the schemes in time and space is proposed in view of increasing the numerical efficiency and reducing the computational time. The method is based on a four-parameter family of schemes we shall tune in function of the physical data (velocity, diffusion), the characteristic size in time and space, and the local regularity of the function leading to a nonlinear procedure. The a posteriori strategy we adopt consists in, given the solution at time t n , computing a candidate solution with the highest accurate schemes in time and space for all the nodes. Then, for the nodes that present some instabilities, both the schemes in time and space are modified and adapted in order to preserve the stability with a large time step. The updated solution is computed with node-dependent schemes both in time and space. For the sake of simplicity, only convection-diffusion problems are addressed as a prototype with a two-parameters five-points finite difference method for the spatial discretisation together with an explicit time two-parameters four-stages Runge–Kutta method. We prove that we manage to obtain an optimal time-step algorithm that produces accurate numerical approximations exempt of non-physical oscillations. [ABSTRACT FROM AUTHOR]

The conventional no-slip boundary condition does not always hold in several fluid flow applications and must be replaced with appropriate slip conditions according to the wall and fluid properties. However, not only slip boundary conditions are still a subject of discussion among fluid dynamicists, but also their numerical treatment is far from being well-developed, particularly in the context of very high-order accurate methods. The complexity of these conditions significantly increases when the boundary is not aligned with the chosen coordinate system and, even more challenging, when the fluid slips along a curved boundary. The present work proposes a simple and efficient numerical treatment of general slip boundary conditions on arbitrary curved boundaries for three-dimensional fluid flow problems governed by the incompressible Navier–Stokes equations. In that regard, two critical challenges arise: (i) achieving very high-order of convergence with arbitrary curved boundaries for the classical no-slip boundary conditions and (ii) extending the developed numerical techniques to impose general slip boundary conditions. The conventional treatment of curved boundaries relies on generating curved meshes to eliminate the geometrical mismatch between the physical and computational boundaries and achieve high-order of convergence. However, such an approach requires sophisticated meshing algorithms, cumbersome quadrature rules on curved elements, and complex non-linear transformations. In contrast, the reconstruction for off-site data (ROD) method handles arbitrary curved boundaries approximated with linear piecewise elements while employing polynomial reconstructions with specific linear constraints to fulfil the prescribed boundary conditions. For that purpose, the general slip boundary conditions are reformulated on a local orthonormal basis to allow a straightforward application of the ROD method with scalar boundary conditions. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed solely on the polygonal mesh elements. Several benchmark test cases of fluid flow problems in non-trivial three-dimensional curved domains are addressed and confirm that the proposed method effectively achieves up to the eighth-order of convergence. • Arbitrary 3D curved geometries approximated with piecewise linear meshes. • Staggered finite volume method for the incompressible Navier-Stokes equations. • Several benchmark test cases with analytical and manufactured solutions. • Eighth-order of convergence for the no-slip and general slip boundary conditions. [ABSTRACT FROM AUTHOR]

We propose a new family of high order accurate finite volume schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a detector chain to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers’, and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations. [ABSTRACT FROM AUTHOR]

We propose a new sixth-order finite volume scheme to solve the bidimensional linear steady-state Stokes problem on staggered unstructured meshes and complex geometries. The method is based on several classes of polynomial reconstructions to accurately evaluate the diffusive fluxes, the pressure gradient, and the velocity divergence. The main difficulty is to handle the div-grad duality to avoid numerical locking and oscillations. A new preconditioning technique based on the construction of a pseudo-inverse matrix is also proposed to dramatically reduce the computational effort. Several numerical simulations are carried out to highlight the performance of the new method. [ABSTRACT FROM AUTHOR]

The numerical solution of the incompressible Navier–Stokes equations raises challenging numerical issues on the development of accurate and robust discretisation techniques due to the div-grad duality. Significant advances have been achieved in the context of high-order of convergence methods, but many questions remain unsolved in the quest of practical, accurate, and efficient approaches. In particular, the discretisation of problems with arbitrary curved boundaries has been the subject of intensive research, where the conventional practice consists in employing curved meshes to avoid accuracy deterioration. The present work proposes a novel approach, the reconstruction for off-site data method, which employs polygonal meshes to approximate arbitrary smooth curved boundaries with linear piecewise elements. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed on the polygonal mesh elements. Boundary conditions are taken into account via polynomial reconstructions with specific linear constraints defined for a set of points on the physical boundary. The very-high order of convergence is preserved without relying on the full curve parametrisation, avoiding the limitations of curved mesh approaches, such as sophisticated mesh generation algorithms, cumbersome quadrature rules, and complex non-linear transformations. A comprehensive verification benchmark is provided, with numerical test cases for several fluid flow problems, to demonstrate the capability of the proposed approach to achieve very high-orders of convergence. • Steady-state incompressible Navier–Stokes equations in curved domains. • Very high-order accurate finite volume scheme. • Arbitrary curved boundaries approximated with unstructured polygonal meshes. • Verification benchmark comprising several fluid flows problems. • Sixth-order of convergence for non-trivial arbitrary curved boundaries. [ABSTRACT FROM AUTHOR]

The design of efficient, simple, and easy to code, second-order finite volume methods is an important challenge to solve practical problems in physics and in engineering where complex and very accurate techniques are not required. We propose an extension of the original Frink’s approach based on a cell-to-vertex interpolation to compute vertex values with neighbour cell values. We also design a specific scheme which enables to use whatever collocation point we want in the cells to overcome the mass centre point restrictive choice. The method is proposed for two- and three-dimensional geometries and a second-order extension time-discretization is given for time-dependent equation. A large number of numerical simulations are carried out to highlight the performance of the new method. [ABSTRACT FROM AUTHOR]

During typesetting, Figs. 8 and 21 got corrupted and the images shown in the online published version are not correct. The original publication was updated. [ABSTRACT FROM AUTHOR]

Accuracy may be dramatically reduced when the boundary domain is curved and numerical schemes require a specific treatment of the boundary condition to preserve the optimal order. In the finite volume context, Ollivier-Gooch and Van Altena (2002) has proposed a technique to overcome such limitation and restore the very high-order accuracy which consists in specific restrictions considered in the least-squares minimization associated to the polynomial reconstruction. The method suffers from several drawbacks, particularly, the use of curved elements that requires sophisticated meshing algorithms. We propose a new method where the physical domain and the computational domain are distinct and we introduce the Reconstruction for Off-site Data (ROD) where polynomial reconstructions are carried out on the mesh using data localized outside of the computational domain, namely the Dirichlet condition situated on the physical domain. A series of numerical tests assess the accuracy, convergence rates, robustness, and efficiency of the new method and show that the boundary condition is fully integrated in the scheme with a very high-order accuracy and the optimal convergence rate is achieved. [ABSTRACT FROM AUTHOR]

Breast reduction is one of the most common procedures in breast surgery. The aim of this work is to develop a computational model allowing one to forecast the final breast geometry according to the incision marking parameters. This model can be used in surgery simulators that provide preoperative planning and training, allowing the study of the origin of the errors in breast reduction. From the mathematical point of view, this is a problem of calculus of variations with unusual boundary conditions, known as knitting conditions. The breast tissue is considered as a hyperelastic material, discretized with three-dimensional finite elements for the body, whereas the skin is modelled with two-dimensional finite elements on the curved surface. Although the model is of low precision, we show that it is sufficient for a satisfactory prediction of breast reduction surgery results, allowing an analysis of errors frequently performed during the surgery and giving an understanding of how to avoid or correct them. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

• Conjugate heat transfer problems in arbitrary smooth curved domains. • Very high-order accurate finite volume scheme with polyhedral meshes. • Explicit matrix formulation replacing the standard residual operator formulation. • Error convergence analysis emphasising at least the sixth-order of convergence. • Substantial computational gains for the execution time and memory usage. The conjugate heat transfer problem is found in many engineering applications and comprises the heat transfer between solids and fluids in contact. The discretization method of these problems must guarantee that the interface conditions are properly satisfied, which is an additional difficulty when designing a numerical scheme with very high-order of convergence. The three-dimensional case with arbitrary curved domains is particularly challenging to treat, as the standard approaches to preserve the optimal convergence order become cumbersome with the use of curved meshes and non-linear transformations. Moreover, the implementation aspects of very high-order of convergence methods are critical for the three-dimensional case, as the computational cost of conventional algorithms often grows exponentially with the problem dimension. A very high-order accurate finite volume scheme with general polyhedral meshes is proposed in this work, which is able to solve three-dimensional conjugate heat transfer problems in arbitrary curved domains. The implementation of the proposed method is addressed and optimized calculations are derived to provide the same approximate solution at a significantly reduced computational cost. The provided set of numerical benchmarks, addressing several situations of the conjugate heat transfer problem, allows confirming that the optimal convergence order is effectively achieved. The results obtained also show that the proposed method overcomes the main shortcomings of the standard curved mesh approaches, while preserving the accuracy and the convergence order with the sole use of polyhedral meshes. A computational benchmark proves that substantial performance improvements are obtained from the proposed optimization, which represents a significant advance towards less demanding and more efficient numerical simulations. [ABSTRACT FROM AUTHOR]

The conjugate heat transfer problem is found in non-isothermal physical systems that involve thermodynamic processes between materials that are thermally coupled through non-adiabatic contacts. A very high-order accurate finite volume scheme in general polygonal meshes is proposed to solve conjugate heat transfer problems with arbitrary curved interfaces and imperfect thermal contacts. Besides the discontinuous thermal properties, imperfect thermal contacts are challenging to address since the obtained temperature is also discontinuous on the interface as a consequence of the interfacial thermal resistance. Moreover, the arbitrary curved interfaces are discretized with polygonal meshes to avoid the shortcomings of classical curved mesh approaches, but a specific treatment is required to overcome the geometrical mismatch and properly fulfill the prescribed interface conditions. The proposed method is based on a partitioned formulation of the conjugate heat transfer problem with the appropriate thermal coupling. A generic polynomial reconstruction method is used to provide local approximations of the temperature complemented with the reconstruction for off-site data method to properly fulfill the prescribed interface conditions. A comprehensive numerical benchmark is provided to verify the proposed method and assess its capability in terms of accuracy, convergence order, stability, and robustness. The results provide the optimal very high-order of convergence and prove the capability of the method to handle arbitrary curved interfaces and imperfect thermal contacts. This contribution represents a significant step towards more efficient and versatile numerical methods for complex conjugate heat transfer problems in engineering applications. [ABSTRACT FROM AUTHOR]