arXiv subjects

Maurizio Tavelli

Publications and source records attributed to Maurizio Tavelli.

At least 19 recordsLinked to original sources

arXiv2026

On the treatment of topology changes on 3D polyhedral moving meshes via 4D space-time hole-like elements in direct ALE ADER-DG methods

This work investigates a novel approach for the high order evolution of hyperbolic PDEs using ADER discontinuous Galerkin schemes within a direct Arbitrary-Lagrangian-Eulerian (ALE) framework on 3D moving polyhedral meshes with topology changes. Our direct ALE method is based on the PDE integration over 4D (3D+time) space-time control volumes connecting the elements of two subsequent tessellations, so to simultaneously evolve the solution both in time and between the two different meshes in an effective and high order manner. In this way, we also avoid any complex and expensive projection-reconstruction techniques and any mesh intersection operation typical of indirect ALE schemes. The crucial step consists in the strategy for building space-time control volumes that also connect elements with different shapes and neighborhoods due to a change in topology. In fact, simply linking existing elements by collapsing or expanding their edges would leave a "hole" in the space-time domain. To fill it, we introduce additional degenerate elements that we call hole-like elements. These are 4D objects with zero 3D volume at both the beginning and end of the timestep, but which possess a strictly non-zero 4D space-time volume. Given the uniqueness of this space-time approach in 3D+time and the necessity of characterizing the geometry of such elements, the main objective of this paper is the formal geometrical and numerical description of the method as well as the presentation of new and intuitive visualization strategies. In particular, we provide a detailed characterization of the hole-like elements arising in correspondence to 2-3, 3-2, and 4-4 flips on the underlying Delaunay tetrahedralization. Finally, we numerically show that the method is fully conservative, satisfies the GCL and maintains the correct order of convergence even in the presence of frequent topology changes.

math.NA↗

arXiv2026

A semi-implicit two dimensional solver for a covariant formulation of the shallow water equations

In this paper we combine a flexible covariant formulation of the shallow water equations with the semi-implicit numerical scheme developed over the years by Casulli and collaborators. After adopting an orthogonal, but non-orthonormal, coordinate basis on two dimensional manifolds, and by writing the divergence of symmetric tensors in a way that avoids the introduction of Christoffel symbols, the shallow water equations preserve a very close resemblance to the usual one expressed in Cartesian coordinates. In this way, a stable semi-implicit scheme can be derived by using an implicit discretization for the gradient of surface elevation in the momentum equations and for the velocity in the continuity equation, with stability properties that are independent of the celerity. We have tested the new method over a variety of challenging benchmarks, including, among the others, the smooth wave propagation over a water globe and the deformation of an artery branch. Two appealing additional features make the method particularly powerful with respect to oceanographic applications: firstly, thanks to the wetting and drying ability of our semi-implicit approach, no pathological behaviors occur at the poles; secondly, the scheme is naturally well-balanced, and it is able to preserve perfect stationarity, up to machined precision, of the entire ocean configuration of the earth.

physics.flu-dyn↗

arXiv2026

A structure-preserving semi-implicit four-split scheme for continuum mechanics

We introduce a novel structure-preserving vertex-staggered semi-implicit four-split discretization of a unified first order hyperbolic formulation of continuum mechanics that is able to describe at the same time fluid and solid materials within the same mathematical model. The governing PDE system goes back to pioneering work of Godunov, Romenski, Peshkov and collaborators. Previous structure-preserving discretizations of this system allowed to respect the curl-free properties of the distortion field and the specific thermal impulse in the absence of source terms and were consistent with the low Mach number limit with respect to the adiabatic sound speed. However, the evolution of the thermal impulse and the distortion field were still discretized explicitly, thus requiring a rather severe CFL stability restriction on the time step based on the shear sound speed and the finite, but potentially large, speed of heat waves. Instead, the new four-split semi-implicit scheme presented in this paper has a material time step restriction only. For this purpose, the governing PDE system is split into four subsystems: i) a convective subsystem, which is the only one that is treated explicitly; ii) a heat subsystem, iii) a subsystem containing momentum, distortion field and specific thermal impulse; iv) a pressure subsystem. The three subsystems ii)-iv) are all discretized implicitly, hence a rather mild CFL restriction based on the velocity of the continuum is imposed. The method is asymptotically consistent with the low Mach number limit and the stiff relaxation limits. Moreover, it maintains an exactly curl-free distortion field and thermal impulse in the case of linear source terms or in their absence. The scheme is benchmarked against classical test cases verifying its theoretical properties.

math.NA↗

arXiv2025

SCOUT: Semi-Lagrangian COnservative and Unconditionally sTable schemes for nonlinear advection-diffusion problems

In this work, we propose a new semi-Lagrangian (SL) finite difference scheme for nonlinear advection-diffusion problems. To ensure conservation, which is fundamental for achieving physically consistent solutions, the governing equations are integrated over a space-time control volume constructed along the characteristic curves originating from each computational point. By applying Gauss theorem, all space-time surface integrals can be evaluated. For nonlinear problems, a nonlinear equation must be solved to find the foot of the characteristic, while this is not needed in linear cases. This formulation yields SL schemes that are fully conservative and unconditionally stable, as verified by numerical experiments with CFL numbers up to 100. Moreover, the diffusion terms are, for the first time, directly incorporated within a conservative semi-Lagrangian framework, leading to the development of a novel characteristic-based Crank-Nicolson discretization in which the diffusion contribution is implicitly evaluated at the foot of the characteristic. A broad set of benchmark tests demonstrates the accuracy, robustness, and strict conservation property of the proposed method, as well as its unconditional stability.

math.NA↗

arXiv2023

A high order accurate space-time trajectory reconstruction technique for quantitative particle trafficking analysis

The study of moving particles (e.g. molecules, virus, vesicles, organelles, or whole cells) is crucial to decipher a plethora of cellular mechanisms within physiological and pathological conditions. Powerful live-imaging approaches enable life scientists to capture particle movements at different scale from cells to single molecules, that are collected in a series of frames. However, although these events can be captured, an accurate quantitative analysis of live-imaging experiments still remains a challenge. Two main approaches are currently used to study particle kinematics: kymographs, which are graphical representation of spatial motion over time, and single particle tracking (SPT) followed by linear linking. Both kymograph and SPT apply a space-time approximation in quantifying particle kinematics, considering the velocity constant either over several frames or between consecutive frames, respectively. Thus, both approaches intrinsically limit the analysis of complex motions with rapid changes in velocity. Therefore, we design, implement and validate a novel reconstruction algorithm aiming at supporting tracking particle trafficking analysis with mathematical foundations. Our method is based on polynomial reconstruction of 4D (3D+time) particle trajectories, enabling to assess particle instantaneous velocity and acceleration, at any time, over the entire trajectory. Here, the new algorithm is compared to state-of-the-art SPT followed by linear linking, demonstrating an increased accuracy in quantifying particle kinematics. Our approach is directly derived from the governing equations of motion, thus it arises from physical principles and, as such, it is a versatile and reliable numerical method for accurate particle kinematics analysis which can be applied to any live-imaging experiment where the space-time coordinates can be retrieved.

math.NA↗

arXiv2022

An all Froude high order IMEX scheme for the shallow water equations on unstructured Voronoi meshes

We propose a novel numerical method for the solution of the shallow water equations in different regimes of the Froude number making use of general polygonal meshes. The fluxes of the governing equations are split such that advection and acoustic-gravity sub-systems are derived, hence separating slow and fast phenomena. This splitting allows the nonlinear convective fluxes to be discretized explicitly in time, while retaining an implicit time marching for the acoustic-gravity terms. Consequently, the novel schemes are particularly well suited in the low Froude limit of the model, since no numerical viscosity is added in the implicit solver. Besides, stability follows from a milder CFL condition which is based only on the advection speed and not on the celerity. High order time accuracy is achieved using the family of semi-implicit IMEX Runge-Kutta schemes, while high order in space is granted relying on two discretizations: (i) a cell-centered finite volume (FV) scheme for the nonlinear convective contribution on the polygonal cells; (ii) a staggered discontinuous Galerkin (DG) scheme for the solution of the linear system associated to the implicit discretization of the pressure sub-system. Therefore, three different meshes are used, namely a polygonal Voronoi mesh, a triangular subgrid and a staggered quadrilateral subgrid. The novel schemes are proved to be Asymptotic Preserving (AP), hence a consistent discretization of the limit model is retrieved for vanishing Froude numbers, which is the given by the so-called "lake at rest" equations. Furthermore, the novel methods are well-balanced by construction, and this property is also demonstrated. Accuracy and robustness are then validated against a set of benchmark test cases with Froude numbers ranging in the interval $\Fr \approx [10^{-6};5]$, hence showing that multiple time scales can be handled by the novel methods.

math.NA↗

arXiv2021

On the construction of conservative semi-Lagrangian IMEX advection schemes for multiscale time dependent PDEs

This article is devoted to the construction of a new class of semi-Lagrangian (SL) schemes with implicit-explicit (IMEX) Runge-Kutta (RK) time stepping for PDEs involving multiple space-time scales. The semi-Lagrangian (SL) approach fully couples the space and time discretization, thus making the use of RK strategies particularly difficult to be combined with. First, a simple scalar advection-diffusion equation is considered as a prototype PDE for the development of a high order formulation of the semi-Lagrangian IMEX algorithms. The advection part of the PDE is discretized explicitly at the aid of a SL technique, while an implicit discretization is employed for the diffusion terms. Second, the SL-IMEX approach is extended to deal with hyperbolic systems with multiple scales, including balance laws, that involve shock waves and other discontinuities. A novel SL technique is proposed, which is based on the integration of the governing equations over the space-time control volume which arises from the motion of each grid point. High order of accuracy is ensured by the usage of IMEX RK schemes combined with a Cauchy-Kowalevskaya procedure that provides a predictor solution within each space-time element. The one-dimensional shallow water equations (SWE) are chosen to validate the new conservative SL-IMEX schemes, where convection and pressure fluxes are treated explicitly and implicitly, respectively. The asymptotic-preserving (AP) property of the novel schemes is also studied considering a relaxation PDE system for the SWE. A large suite of convergence studies for both the non-conservative and the conservative version of the novel class of methods demonstrates that the formal order of accuracy is achieved and numerical evidences about the conservation property are shown. The AP property for the corresponding relaxation system is also investigated.

math.NA↗

arXiv2020

A unified first order hyperbolic model for nonlinear dynamic rupture processes in diffuse fracture zones

Earthquake fault zones are more complex, both geometrically and rheologically, than an idealised infinitely thin plane embedded in linear elastic material. To incorporate nonlinear material behaviour, natural complexities, and multi-physics coupling within and outside of fault zones, here we present a first-order hyperbolic and thermodynamically compatible mathematical model for a continuum in a gravitational field which provides a unified description of nonlinear elasto-plasticity, material damage and of viscous Newtonian flows with phase transition between solid and liquid phases. The fault geometry and secondary cracks are described via a scalar function $ξ\in [0,1]$ that indicates the local level of material damage. The model also permits the representation of arbitrarily complex geometries via a diffuse interface approach based on the solid volume fraction function $α\in [0,1]$. Neither of the two scalar fields $ξ$ and $α$ needs to be mesh-aligned, allowing thus faults and cracks with complex topology and the use of adaptive Cartesian meshes (AMR). The model shares common features with phase-field approaches but substantially extends them. We show a wide range of numerical applications that are relevant for dynamic earthquake rupture in fault zones, including the co-seismic generation of secondary off-fault shear cracks, tensile rock fracture in the Brazilian disc test, as well as a natural convection problem in molten rock-like material.

physics.geo-ph↗

arXiv2020

Space-time adaptive ADER discontinuous Galerkin schemes for nonlinear hyperelasticity with material failure

We are concerned with the numerical solution of a unified first order hyperbolic formulation of continuum mechanics that originates from the work of Godunov, Peshkov and Romenski (GPR model) and which is an extension of nonlinear hyperelasticity that is able to describe simultaneously nonlinear elasto-plastic solids at large strain, as well as viscous and ideal fluids. The proposed governing PDE system also contains the effect of heat conduction and can be shown to be symmetric and thermodynamically compatible. In this paper we extend the GPR model to the simulation of nonlinear dynamic rupture processes and material fatigue effects, by adding a new scalar variable to the governing PDE system. This extra parameter describes the material damage and is governed by an advection-reaction equation, where the stiff and highly nonlinear reaction mechanisms depend on the ratio of the local von Mises stress to the yield stress of the material. The stiff reaction mechanisms are integrated in time via an efficient exponential time integrator. Due to the multiple space-time scales, the model is solved on space-time adaptive Cartesian meshes using high order discontinuous Galerkin finite element schemes with a posteriori subcell finite volume limiting. A key feature of our new model is the use of a twofold diffuse interface approach that allows the cracks to form anywhere and at any time, independently of the chosen computational grid, without requiring that the geometry of the rupture fault be known a priori. We furthermore make use of a scalar volume fraction function that indicates whether a given point is inside the solid or outside, allowing the description of solids of arbitrarily complex shape.

math.NA↗

arXiv2020

ExaHyPE: An Engine for Parallel Dynamically Adaptive Simulations of Wave Problems

ExaHyPE ("An Exascale Hyperbolic PDE Engine") is a software engine for solving systems of first-order hyperbolic partial differential equations (PDEs). Hyperbolic PDEs are typically derived from the conservation laws of physics and are useful in a wide range of application areas. Applications powered by ExaHyPE can be run on a student's laptop, but are also able to exploit thousands of processor cores on state-of-the-art supercomputers. The engine is able to dynamically increase the accuracy of the simulation using adaptive mesh refinement where required. Due to the robustness and shock capturing abilities of ExaHyPE's numerical methods, users of the engine can simulate linear and non-linear hyperbolic PDEs with very high accuracy. Users can tailor the engine to their particular PDE by specifying evolved quantities, fluxes, and source terms. A complete simulation code for a new hyperbolic PDE can often be realised within a few hours - a task that, traditionally, can take weeks, months, often years for researchers starting from scratch. In this paper, we showcase ExaHyPE's workflow and capabilities through real-world scenarios from our two main application areas: seismology and astrophysics.

cs.MS↗

arXiv2020

A novel staggered semi-implicit space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations

A new high order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions. The staggered DG scheme defines the discrete pressure on the primal triangular mesh, while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure finite elements is proposed. On the primary triangular mesh (the pressure elements) the basis functions are piecewise polynomials of degree $N$ and are allowed to jump on the boundaries of each triangle. On the dual mesh instead (the velocity elements), the basis functions consist in the union of piecewise polynomials of degree $N$ on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries, while they are continuous inside. In other words, the basis functions on the dual mesh are built by continuous finite elements on the subtriangles. This choice allows the construction of an efficient, quadrature-free and memory saving algorithm. In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations, arbitrary high order of accuracy in time is achieved through the use of time-dependent test and basis functions, in combination with simple and efficient Picard iterations. Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes, not only from a computer memory point of view, but also concerning the computational time.

math.NA↗

arXiv2019

Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems

We propose a new family of high order staggered semi-implicit discontinuous Galerkin (DG) methods for the simulation of natural convection problems. Assuming small temperature fluctuations, the Boussinesq approximation is valid and the flow can simply be modeled by the incompressible Navier-Stokes equations coupled with a transport equation for the temperature and a buoyancy source term in the momentum equation. Our numerical scheme is developed starting from the work presented in [TD14], in which the spatial domain is discretized using a face-based staggered unstructured mesh. For the computation of the advection and diffusion terms, two different algorithms are presented: i) a purely Eulerian explicit upwind-type scheme and ii) a semi-Lagrangian approach. The first methodology leads to a conservative scheme whose major drawback is the time step restriction imposed by the CFL stability condition. On the contrary, computational efficiency can be notably improved relying on a semi-Lagrangian approach. This method leads to an unconditionally stable scheme if the diffusive terms are discretized implicitly. Once the advection and diffusion contributions have been computed, the pressure Poisson equation is solved and the velocity is updated. As a second model for the computation of buoyancy-driven flows, we also consider the full compressible Navier-Stokes equations. The staggered semi-implicit DG method first proposed in [TD17] for all Mach number flows is properly extended to account for the gravity source terms arising in the momentum and energy conservation laws. The validity and the robustness of our novel class of staggered semi-implicit DG schemes is assessed at the aid of several classical benchmark problems, showing in all cases a good agreement with available numerical reference data. Finally, a detailed comparison between the incompressible and the compressible solver is presented.

physics.comp-ph↗

arXiv2018

Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine

In this paper we discuss a new and very efficient implementation of high order accurate ADER discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication avoiding and cache blocking and are furthermore very well-suited for vectorization, so that they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the numerical algorithm and show some applications to a set of hyperbolic equations with increasing level of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes.

math.NA↗

arXiv2018

A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

In most classical approaches of computational geophysics for seismic wave propagation problems, complex surface topography is either accounted for by boundary-fitted unstructured meshes, or, where possible, by mapping the complex computational domain from physical space to a topologically simple domain in a reference coordinate system. In this paper we propose a completely different strategy. We address the problem of geometrically complex free surface boundary conditions with a novel diffuse interface method on adaptive Cartesian meshes that consists in the introduction of a characteristic function $ 0\leqα\leq 1$ which identifies the location of the solid medium and the surrounding air and thus implicitly defines the location of the free surface boundary. Our new approach completely avoids the problem of mesh generation, since all that is needed for the definition of the complex surface topography is to set a scalar color function to unity inside the regions covered by the solid and to zero outside. An analysis of the eigenvalues of the PDE system shows that the complexity of the geometry has no influence on the admissible time step size due to the CFL condition. The model reduces to the classical linear elasticity equations inside the solid medium where the gradients of $α$ are zero, while in the diffuse interface zone at the free surface boundary the governing PDE system becomes nonlinear. We can prove that the solution of the Riemann problem with arbitrary data and a jump in $α$ from unity to zero yields a Godunov-state at the interface that satisfies the free-surface boundary condition exactly. In order to reduce numerical dissipation, we use high order DG finite element schemes on adaptive AMR grids together with a high resolution shock capturing subcell finite volume (FV) limiter in the diffuse interface region.

math.NA↗

arXiv2018

Arbitrary high order accurate space-time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity

In this paper we propose a new high order accurate space-time DG finite element scheme for the solution of the linear elastic wave equations in first order velocity-stress formulation in two and three-space dimensions on staggered unstructured triangular and tetrahedral meshes. The method reaches arbitrary high order of accuracy in both space and time via the use of space-time basis and test functions. Within the staggered mesh formulation, we define the discrete velocity field in the control volumes of a primary mesh, while the discrete stress tensor is defined on a face-based staggered dual mesh. The space-time DG formulation leads to an implicit scheme that requires the solution of a linear system for the unknown degrees of freedom at the new time level. The number of unknowns is reduced at the aid of the Schur complement, so that in the end only a linear system for the degrees of freedom of the velocity field needs to be solved. Thanks to the use of a spatially staggered mesh, the stencil of the final velocity system involves only the element and its direct neighbors. The resulting linear system can be efficiently solved via matrix-free iterative methods. The chosen discretization and the linear nature of the governing PDE system lead to an unconditionally stable scheme, which allows large time steps even for low quality meshes that contain sliver elements. The fully discrete staggered space-time DG method is proven to be energy stable for any order of accuracy, for any mesh and for any time step size. For the particular case of Crank-Nicolson time discretization and homogeneous material, the final velocity system can be proven to be symmetric and positive definite and in this case the scheme is also exactly energy preserving. The new scheme is applied to several test problems in two and three space dimensions, providing also a comparison with high order explicit ADER-DG schemes

math.NA↗

arXiv2018

A divergence-free semi-implicit finite volume scheme for ideal, viscous and resistive magnetohydrodynamics

In this paper we present a novel pressure-based semi-implicit finite volume solver for the equations of compressible ideal, viscous and resistive magnetohydrodynamics (MHD). The new method is conservative for mass, momentum and total energy and in multiple space dimensions it is constructed in such a way as to respect the divergence-free condition of the magnetic field exactly, also in the presence of resistive effects. This is possible via the use of multi-dimensional Riemann solvers on an appropriately staggered grid for the time evolution of the magnetic field and a double curl formulation of the resistive terms. The new semi-implicit method for the MHD equations proposed here discretizes all terms related to the pressure in the momentum equation and the total energy equation implicitly, making again use of a properly staggered grid for pressure and velocity. The time step of the scheme is restricted by a CFL condition based only on the fluid velocity and the Alfvén wave speed and is not based on the speed of the magnetosonic waves. Our new method is particularly well-suited for low Mach number flows and for the incompressible limit of the MHD equations, for which it is well-known that explicit density-based Godunov-type finite volume solvers become increasingly inefficient and inaccurate due to the increasingly stringent CFL condition and the wrong scaling of the numerical viscosity in the incompressible limit. We show a relevant MHD test problem in the low Mach number regime where the new semi-implicit algorithm is a factor of 50 faster than a traditional explicit finite volume method, which is a very significant gain in terms of computational efficiency. However, our numerical results confirm that our new method performs well also for classical MHD test cases with strong shocks. In this sense our new scheme is a true all Mach number flow solver.

math.NA↗

arXiv2016

A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers

We propose a new arbitrary high order accurate semi-implicit space-time discontinuous Galerkin (DG) method for the solution of the two and three dimensional compressible Euler and Navier-Stokes equations on staggered unstructured curved meshes. The method is pressure-based and semi-implicit and is able to deal with all Mach number flows. In our scheme, the discrete pressure is defined on the primal grid, while the discrete velocity field and the density are defined on a face-based staggered dual grid. All convective terms are discretized explicitly, while the pressure terms appearing in the momentum and energy equation are discretized implicitly. Substitution of the momentum equation into the energy equation yields a linear system for the scalar pressure as the only unknown. The enthalpy and the kinetic energy are taken explicitly and are then updated using a simple Picard procedure. Thanks to the use of a staggered grid, the final pressure system is a very sparse block five-point system for three dimensional problems. The viscous terms and the heat flux are also discretized making use of the staggered grid by defining the viscous stress tensor and the heat flux vector on the dual grid, which corresponds to the use of a lifting operator on the dual grid. The time step of our new numerical method is limited by a CFL condition based only on the fluid velocity and not on the sound speed. This makes the method particularly interesting for low Mach number flows. Finally, a very simple combination of artificial viscosity and the a posteriori MOOD technique allows to deal with shock waves and thus permits also to simulate high Mach number flows. We show computational results for a large set of two and three-dimensional benchmark problems, including both low and high Mach number flows and using polynomial approximation degrees up to p=4.

math.NA↗

arXiv2016

A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes

In this paper we propose a novel arbitrary high order accurate semi-implicit space-time DG method for the solution of the three-dimensional incompressible Navier-Stokes equations on staggered unstructured curved tetrahedral meshes. As typical for space-time DG schemes, the discrete solution is represented in terms of space-time basis functions. This allows to achieve very high order of accuracy also in time, which is not easy to obtain for the incompressible Navier-Stokes equations. Similar to staggered finite difference schemes, in our approach the discrete pressure is defined on the primary tetrahedral grid, while the discrete velocity is defined on a face-based staggered dual grid. A very simple and efficient Picard iteration is used in order to derive a space-time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. From numerical experiments we find that the linear system seems to be reasonably well conditioned, since all simulations shown in this paper could be run without the use of any preconditioner. For a piecewise constant polynomial approximation in time and proper boundary conditions, the resulting system is symmetric and positive definite. This allows us to use even faster iterative solvers, like the conjugate gradient method. The proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3D test problems and by comparing the obtained numerical results with available exact analytical solutions, or with other numerical or experimental reference data.

math.NA↗