Adaptive Cartesian Mesh Research Articles

Prediction–correction projection immersed interface method for interfacial stress-induced flows and applications to electrohydrodynamics

We propose a robust, high-resolution prediction–correction projection immersed interface method (IIM) for solving the unsteady incompressible Navier–Stokes equations with traction boundary conditions, which arise from free surface flows driven by capillary, electric, and elastic forces. This method combines the advantages of traditional body-fitted moving mesh methods and immersed boundary methods (IBM), allowing for the accurate imposition of boundary conditions on free surfaces using Cartesian grids and providing detailed interface information that is typically smoothed out in traditional IBM. The irregular liquid-phase domain is embedded within a square region and discretized using a dynamically adaptive Cartesian mesh. The free surface is captured using a narrow-band level set with hybrid reinitialization. A prediction–correction projection scheme is constructed, incorporating additional pressure predictions and corrections to enhance robustness. The resulting Helmholtz/Poisson equations are solved using the augmented IIM for boundary value problems. Grid refinement analysis demonstrates second-order convergence of the L∞ error, even in challenging cases such as the oscillating drop test with low viscosity. We further apply this method to electrohydrodynamic (EHD) problems, constructing an implicit augmented IIM to solve the equations governing the electric field and charge conservation. Numerical experiments demonstrate that this method accurately addresses highly intense EHD phenomena, such as the formation of Taylor cones, highlighting its robustness.

Accurate numerical prototypes of microfluidic droplet generators with open source tools

Droplet-based microfluidics gained significant attention for its high technological impact in various fields like (bio)analysis and (bio)synthesis. Precise and controlled droplet size is critical, for the encapsulated products, or the yield of chemical reactions. In a broad range of experimental parameters, the understanding of how droplets form, interact and move with accurate predictive models is crucial. In this work, numerical prototypes of droplet generators were made with Basilisk, an open source software for solving partial differential equations on adaptive Cartesian meshes including grid adaptation and scalability for High-Performance Computing (HPC). This research aims to analyze and compare the obtained droplets against existing experimental data. The evaluation involves qualitative and quantitative comparisons, considering various channel geometries, flow rates, and rheological conditions. The validation of the proposed tool in terms of accuracy and computational performance, enable us to offer to the microfluidics community a reliable tool to design and optimize droplet generators.

Stochastic and self-consistent 3D modeling of streamer discharge trees with Kinetic Monte Carlo

This paper contains the foundation for a new Particle-In-Cell model for gas discharges, based on Îto diffusion and Kinetic Monte Carlo (KMC). In the new model the electrons are described with a microscopic drift-diffusion model rather than a macroscopic one. We discuss the connection of the Îto-KMC model to the equations of fluctuating hydrodynamics and the advection-diffusion-reaction equation which is conventionally used for simulating streamer discharges. The new model is coupled to a particle description of photoionization, providing a non-kinetic all-particle method with several attractive properties, such as: 1) Taking the same input as a fluid model, e.g. mobility coefficients, diffusion coefficients, and reaction rates. 2) Guaranteed non-negative densities. 3) Intrinsic support for reactive and diffusive fluctuations. 4) Exceptional stability properties. The model is implemented as a particle-mesh model on cut-cell grids with Cartesian adaptive mesh refinement. Positive streamer discharges in atmospheric air are considered as the primary application example, and we demonstrate that we can self-consistently simulate large discharge trees.

An immersed selective discontinuous Galerkin method in particle-in-cell simulation with adaptive Cartesian mesh and polynomial preserving recovery

In this paper, a selective discontinuous Galerkin (SDG) method and a polynomial preserving recovery (PPR) method are developed and integrated with the immersed-finite-element particle-in-cell (IFE-PIC) method, in order to carry out the plasma-material interaction simulation on adaptive Cartesian meshes (ACM). To significantly save the computational cost in practice, the PIC simulation often needs to use Cartesian meshes for the whole domain and the ACM for some focused regions in the plasma-material interaction problems. Therefore, we utilize the SDG method with immersed finite elements for the field solver of the IFE-PIC method, as the key technique to allow the local Cartesian mesh refinement which will generate hanging nodes. To minimize the degree of freedom of the SDG on the ACM and reduce the computational cost, a new selective technique of the discontinuous global basis functions is proposed for the SDG. Various types of nodes in the ACM are discussed for the implementation of this selective technique, including the hanging nodes. Meanwhile, the gathering and scattering steps of the PIC simulation also need to be appropriately performed on the ACM. The PPR method is a generic gradient recovery technique to be incorporated into the IFE-PIC method for more accurate force deposition on the ACM. In addition, two other algorithmic structures of the traditional IFE-PIC method have been modified to accommodate the ACM in the simulation, including a new mapping array structure for the particle positioning and a new charge deposition with some special particle positions in the ACM. As a result, the integrated immersed-selective-discontinuous-Galerkin particle-in-cell (ISDG-PIC) method can efficiently perform the plasma-material interactions simulation on the Cartesian meshes with local mesh refinement independent of the interface. Several numerical experiments are performed to demonstrate the convergence, efficiency, and applicability of the proposed ISDG-PIC method.

A Quantitative Approach for the Accurate CFD Simulation of Hover in Turbulent Flow

Time-dependent Navier–Stokes simulations have been carried out for a V22 rotor in hover using an improved, lowdissipation, HLLE ++ upwind algorithm in the OVERFLOW code. Emphasis is placed on lessons learned over the past decade regarding the effects of high-order spatial accuracy, grid resolution, and the use of detached eddy simulation in predicting the figure-of-merit, the rotor's chief performance parameter. A general quick-start procedure is described together with a statistical measure of FM convergence that reduces hover computations by fourfold, similar to computational work for forward flight. Cartesian adaptive mesh refinement is used to resolve the tip–vortex to its correct physical size. This adaptive mesh refinement in the rotor wake also revealed a complex turbulent flow with worm-like structures of various scales. These structures were numerically found a decade ago and recently observed in experiment.

Numerical modelling of hook and claw-type vacuum pump performance based on cut cell Cartesian method

Vacuum pump is widely used in aviation, chemical, semiconductor and other fields because of its superior performance. In order to study the internal flow characteristics and pump performance, 1-D Chamber Model method and the CFD modelling method based on static grid are commonly used, such methods ignore many important three-dimensional transient flow details. However, in numerical modelling of three-dimensional transient flow field for hook and claw-type vacuum pump with cusps in the rotor profile, the generation of flow field grid is always a challenge. The body-fitted mesh method is difficult to generate effective dynamic grid for such kind of fluid domain, which usually leads to the divergence of calculation. In order to solve these problems, the cut cell Cartesian method is proposed to simulate the three-dimensional transient flow field in hook and claw-type vacuum pump. Firstly, the mathematical basis of the cut cell Cartesian method was introduced in detail, in which the immersed boundary method (IBM) was combined with the generation process of adaptive Cartesian volume mesh. Aiming at modelling the hook and claw-type vacuum pump, firstly, the profile design method of hook and claw-type vacuum pump was derived, and the geometric model of the calculation domain was established. The transient flow field characteristics of the vacuum pump with different inter-lobe clearance sizes were analyzed, and the velocity, pressure, vortices and temperature characteristics inside the vacuum pump were obtained. The influence of clearance variation on the performance of the vacuum pump was discussed. The feasibility and accuracy of the method were verified by a study case of a two-stroke reciprocating piston pump. The research provides a good reference for dynamic grid generation and precise numerical simulation of hook and claw-type vacuum pump.

A three-dimensional solver for simulating detonation on curvilinear adaptive meshes

A generic solver in a parallel Cartesian adaptive mesh refinement framework is extended to simulate detonations on three-dimensional structured curvilinear meshes. A second-order accurate finite volume method is used with grid-aligned Riemann solvers for thermally perfect gas mixtures. Detailed, multi-step chemical kinetic mechanisms are employed and numerically incorporated with a splitting approach. The adaptive mesh refinement technique is applied to a mapped mesh using modified prolongation and restriction operators. The flux along the coarse-fine interface is considered in a correction procedure to ensure the conservation of the solver. The numerical accuracy, conservation and robustness of the simulations are verified and validated with suitable benchmark tests. The new solver is then used to simulate detonation problems in non-Cartesian geometries. A simulation is conducted of the three-dimensional detonation propagation in a 90-degree pipe bend. A detonation in a round tube is also simulated in a Galilean frame of reference. Both a rectangular mode and a spinning mode are observed in the simulations. In addition, the fundamental problem of detonation wave/boundary layer interaction is studied. The results show that the new solver can simulate high-speed reactive flows efficiently by the combined use of a curvilinear mapping with mesh adaptation.

Fluid-structure interaction simulations of the ASPIRE SR01 supersonic parachute flight test

High-fidelity fluid-structure interaction (FSI) simulations of the ASPIRE SR01 supersonic parachute flight test are performed through the loose coupling of computational fluid dynamics (CFD) and computational structural dynamics (CSD) solvers. The CFD solver employs a Cartesian adaptive mesh refinement (AMR) grid paradigm with a ghost cell immersed boundary method to represent arbitrarily complex geometries, and the CSD solver uses MITC3 triangular shell and Timoshenko beam elements to discretize the parachute canopy and suspension lines. The CFD resolution required to capture the deficit of the dynamic pressure in the wake of the ASPIRE payload and the aerodynamic drag on the parachute canopy is systematically studied via grid refinement study before FSI simulations are conducted. FSI simulations presented consider the inflation of an initially folded parachute canopy in freestream conditions matching the line stretch event during the ASPIRE SR01 flight test. The total pull force at the instance of peak loading is predicted to within 10% of the flight test data. The sensitivity of the total pull force and canopy projected area to the grid resolution in the CFD and CSD domains is systematically studied via another grid refinement study, and grid convergence with respect to the CFD domain resolution is demonstrated. The pull force vs time curve predicted by each CFD and CSD grid resolution varies by less than 1% indicating a high degree of repeatability and low grid-related uncertainty. Remaining differences in the qualitative behavior of the pull force curve and quantitative difference in the peak pull force are thus determined to stem from modeling errors such as the lack of deceleration, and different initial parachute and flow states in the FSI simulations compared to the flight test.

A stable discontinuous Galerkin method for linear elastodynamics in 3D geometrically complex elastic solids using physics based numerical fluxes

Time-stable, high order accurate and explicit numerical methods are effective for hyperbolic wave propagation problems. As a result of the complexities of real geometries, internal interfaces and nonlinear boundary and interface conditions, discontinuities and sharp wave fronts may become fundamental features of the solution. Therefore, geometrically flexible and adaptive numerical algorithms are crucial for high fidelity and efficient simulations of wave phenomena in many applications. Adaptive curvilinear meshes hold promise to minimise the effort to represent complicated geometries or heterogeneous material data avoiding the bottleneck of feature-preserving meshing. To enable the design of stable DG methods on three space dimensional (3D) curvilinear elements we construct a structure preserving skew-symmetric coordinate transformation motivated by the underlying physics. Using a physics-based numerical penalty-flux, we develop a 3D provably energy-stable discontinuous Galerkin finite element approximation of the elastic wave equation in geometrically complex and heterogeneous media. By construction, our numerical flux is upwind and yields a discrete energy estimate analogous to the continuous energy estimate. The ability to treat conforming and non-conforming curvilinear elements allows for flexible adaptive mesh refinement strategies. The numerical scheme has been implemented in ExaHyPE, a simulation engine for parallel dynamically adaptive simulations of wave problems on adaptive Cartesian meshes. We present 3D numerical experiments of wave propagation in heterogeneous isotropic and anisotropic elastic solids demonstrating stability and high order accuracy. We demonstrate the potential of our approach for computational seismology in a regional wave propagation scenario in a geologically constrained 3D model including the geometrically complex free-surface topography of Mount Zugspitze, Germany.

The usage of adaptive mesh refinement in simulation of high-velocity collision between impactor and thin-walled containment

The problem of collision between spherical impactor and thin-walled containment involves geometrical multi-scale – shell thickness and impactor radius, specifically, 2 mm and 6 mm, are of two orders smaller than shell radius equal to 150 mm. Good spatial resolution for impactor and shell is required as their interfaces deformation is an object of investigation. Uniform meshes with a desirable spatial resolution become computationally impractical due to huge cells number. In order to overcome this difficultly we use dynamically adaptive Cartesian meshes. Computational cells can be split or merged during the computational run accordingly to a user's criterion designed to keep good spatial resolution only at sufficient domains of the flow. The usage of adaptive mesh refinement allows to achieve high accuracy of the numerical simulation in a smaller physical time. We discuss some issues on the usage of adaptive meshes for considered impact problem and present load balancing algorithm for distributed memory parallel calculations.

A solver for simulating shock-induced combustion on curvilinear adaptive meshes

A generic solver in a structured Cartesian adaptive mesh refinement framework is extended to simulate unsteady shock-induced combustion problems on a structured curvilinear mesh. A second-order accurate finite volume method is used with a grid-aligned Riemann solver for inviscid thermally perfect gas mixtures. To solve these reactive problems, detailed chemical kinetic mechanisms are employed with a splitting approach. The prolongation and restriction operators are modified to implement the adaptive mesh refinement algorithm on a mapped mesh. The developed solver is verified with several benchmark tests and is then used to simulate unsteady shock-induced combustion. The results show that the computed stand-off distance of waves and oscillation frequencies of mass fraction of products observed at the stagnation point are in good agreement with the results from experiments.

Experimentally constrained multidimensional simulation of laser-generated plasmas and its application to UV nanosecond ablation of Se and Te

We carry out simulations of laser plasmas generated during UV nanosecond pulsed laser ablation of the chalcogens selenium (Se) and tellurium (Te), and compare the results to experiments. We take advantage of a 2D-axisymmetric, adaptive Cartesian mesh framework that enables plume simulations out to centimeter distances over tens of microseconds. Our model and computational technique enable comparison to laser-plasma applications where the long-term behavior of the plume is of primary interest, such as pulsed laser synthesis and modification of materials. An effective plasma absorption term is introduced in the model, allowing the simulation to be constrained by experimental time-of-flight kinetic energy distributions. We show that the effective simulation qualitatively captures the key characteristics of the observed laser plasma, including the effect of laser spot size. Predictions of full-scale experimentally-constrained Se and Te plasmas for 4.0 J cm−2 laser fluence and 1.8 mm2 circular laser spot area show distinct behavior compared to more commonly studied copper (Cu) plumes. The chalcogen plumes have spatial gradients of plasma density that are steeper than those for Cu by up to three orders of magnitude. Their spatial ion distributions have central bulges, in contrast to the edge-only ionization of Cu. For the irradiation conditions explored, the range of plasma temperatures for Se and Te is predicted to be higher than for Cu by more than 0.50 eV.

Implicit and coupled fluid plasma solver with adaptive Cartesian mesh and its applications to non-equilibrium gas discharges

We present a new fluid plasma solver with adaptive Cartesian mesh (ACM) based on a full-Newton (nonlinear, implicit) scheme for non-equilibrium gas discharge plasma. The electrons and ions are described using drift–diffusion approximation coupled to Poisson equation for the electric field. The electron-energy transport equation is solved to account for electron thermal conductivity, Joule heating, and energy loss of electrons in collisions with neutral species. The rate of electron-induced ionization is a function of electron temperature and could also depend on electron density (important for plasma stratification). The ion and gas temperature are kept constant. The transport equations are discretized using a non-isothermal Scharfetter–Gummel scheme to resolve possible large temperature gradients in the sheaths. We demonstrate the new solver for simulations of direct current (DC) and radiofrequency (RF) discharges. The implicit treatment of the coupled equations allows using large time steps. The full-Newton method (FNM) enables fast nonlinear convergence at each time step, offering significantly improved simulation efficiency. We discuss the selection of time steps for solving different plasma problems. The new solver enables solving several problems we could not solve before with existing software: two- and three-dimensional structures of the entire DC discharges including cathode and anode regions, electric field reversals and double-layer formation, the normal cathode spot and an anode ring, moving striations in diffuse and constricted DC discharges, and standing striations in RF discharges. The developed FNM-ACM technique offers many benefits for tackling the disparity of gas discharge plasma systems' time scales and nonlinearity.

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 idealized 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 ∈ [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 ∈ [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.This article is part of the theme issue ‘Fracture dynamics of solid materials: from particles to the globe’.

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

In this work, we consider the general family of the so called ADER P_NP_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 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), the usual Discontinuous Galerkin (DG) methods (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. In all cases with M ge N > 0 the P_NP_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 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 schemes, due to the use of a rather fine subgrid of 2N+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.

Dynamic wetting failure in curtain coating by the Volume-of-Fluid method

In this paper, we investigate dynamic wetting in the curtain coating configuration. The two-phase Navier–Stokes equations are solved by a Volume-of-Fluid method on an adaptive Cartesian mesh. We introduce the Navier boundary condition to regularize the solution at the triple point and remove the implicit numerical slip induced by the cell-centered interface advection. We use a constant contact angle to describe the dynamic contact line. The resolution of the governing equations allows us to predict the substrate velocity at which wetting failure occurs. The model predictions are compared with prior computations of Liu et al. [C.Y. Liu, E. Vandre, M. Carvalho, S. Kumar, J. Fluid Mech. 808, 290 (2016); C.Y. Liu, M. Carvalho, S. Kumar, Chem. Eng. Sci. 195, 74 (2019)] and experimental observations of Blake et al. [T. Blake, M. Bracke, Y. Shikhmurzaev, Phys. Fluids 11, 1995 (1999)] and Marston et al. [J. Marston, V. Hawkins, S. Decent, M. Simmons, Exp. Fluids 46, 549 (2009)].

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 goes back to the work of Godunov, Peshkov and Romenski [65,68,97] (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, as it obeys the first and second law of thermodynamics. In this paper we extend the GPR model to the simulation of nonlinear dynamic rupture processes, which can be achieved by adding an additional scalar 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 equivalent 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 spatial and temporal scales involved in the problem of crack generation and propagation, the model is solved on space–time adaptive Cartesian meshes using high order accurate discontinuous Galerkin finite element schemes endowed with an a posteriori subcell finite volume limiter. 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, which is simply adaptive Cartesian (AMR). This is substantially different from many fracture modeling approaches that need to resolve discontinuities explicitly, such as for example dynamic shear rupture models used in computational seismology, where the geometry of the rupture fault needs to prescribed a priori. We furthermore make use of a scalar volume fraction function α that indicates whether a given spatial point is inside the solid (α=1) or outside (α=0), thus allowing the description of solids of arbitrarily complex shape. We show extensive numerical comparisons with experimental results for stress-strain diagrams of different real materials and for the generation and propagation of fracture in rocks and pyrex glass at low and high velocities. Overall, a very good agreement between numerical simulations and experiments is obtained. The proposed model is also naturally able to describe material fatigue.

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

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 et al. (1995)[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.

Simplified Modeling of Charged Gas Particles Coupled With EM Field Using Monte Carlo and Finite Volume Methods

A simplified numerical model to simulate plasma occurring in a magnetron sputtering device is presented. Motions and collisions of charged gas particles are modeled by the direct simulation Monte Carlo (DSMC) method on an adaptive Cartesian mesh structure. An electromagnetic (EM) field equation is solved by the finite volume method, a discretization method. The potential of the EM field influences the charged gas particles calculated by DSMC through a simplified form of non-Sibson interpolation scheme developed in this article. The DSMC, EM field solver, and interpolation module are validated separately, and integrated calculations are applied to a test scenario in a magnetron sputtering experiment. Although simulation is performed without plasma calculation, the erosion rates obtained from the simulation agreed well with experimental data, with less than 10% difference. The proposed model is expected to be used in real plasma applications, especially for calculations of erosion and deposition rates without rigorous plasma calculation.

High Order ADER Schemes for Continuum Mechanics

In this paper we first review the development of high order ADER finite volume and ADER discontinuous Galerkin schemes on fixed and moving meshes, since their introduction in 1999 by Toro et al. We show the modern variant of ADER based on a space-time predictor-corrector formulation in the context of ADER discontinuous Galerkin schemes with a posteriori subcell finite volume limiter on fixed and moving grids, as well as on space-time adaptive Cartesian AMR meshes. We then present and discuss the unified symmetric hyperbolic and thermodynamically compatible (SHTC) formulation of continuum mechanics developed by Godunov, Peshkov and Romenski (GPR model), which allows to describe fluid and solid mechanics in one single and unified first order hyperbolic system. In order to deal with free surface and moving boundary problems, a simple diffuse interface approach is employed, which is compatible with Eulerian schemes on fixed grids as well as direct Arbitrary-Lagrangian-Eulerian methods on moving meshes. We show some examples of moving boundary problems in fluid and solid mechanics.