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We present an hp-adaptive discretization for a sharp interface model with a level-set ghost-fluid method to simulate compressible multiphase flows. The scheme applies an efficient p-adaptive discontinuous Galerkin (DG) operator in regions of smooth flow. Shocks and the phase interface are captured by a Finite Volume (FV) scheme on a h-refined element-local sub-grid. The resulting hp-adaptive scheme thus combines both the high order accuracy of the DG method and the robustness of the FV scheme by using p-adaptation in smooth areas and h-refinement at discontinuities, respectively. For the level-set based interface tracking, a similar hybrid DG/FV operator is employed. Both p-refinement and FV shock and interface capturing are performed at runtime and controlled by an indicator, which is based on the modal decay of the solution polynomials. In parallel simulations, the hp-adaptive discretization together with the costly interface tracking algorithm cause a significant imbalance in the processor workloads. To ensure parallel efficiency, we propose a dynamic load balancing scheme that determines the workload distribution by element-local wall time measurements and redistributes elements along a space filling curve. The parallelization strategy is supported by strong scaling tests using up to 8192 cores. The framework is applied to established benchmarks problems for inviscid, compressible multiphase flows. The results demonstrate that the hybrid adaptive discretization can efficiently and accurately handle complex multiphase flow problems involving pronounced interface deformations and merging interface contours. [ABSTRACT FROM AUTHOR]

The classical paper [40] by S. K. Godunov had a revolutionary effect on the field of numerical simulations of compressible fluid flows. Yet, for some twenty years after its publication, it was mostly used by the mathematical community in the Soviet Union. The seminal paper of van Leer [85] has inaugurated the period of universal interest in high-resolution extensions of Godunov's scheme. The first step to take was to modify the (locally) self-similar solution to the Riemann Problem (at discontinuities) by allowing piecewise-polynomial (rather than piecewise-constant) initial data. The GRP (Generalized Riemann Problem) analysis [6] provided analytical solutions (for piecewise-linear data) that could be readily implemented in a high-resolution robust code. This review paper focuses on the evolution of some mathematical aspects of this method. It addresses the three concepts of consistency, stability and convergence in the context of compact finite difference (or finite volume) schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of "balance laws", a common viewpoint in relevant physical conservation laws. The first significant observation is that under very mild conditions a weak solution is indeed a solution to the balance law (obtained by a formal application of the Gauss-Green formula). Since high-resolution schemes require the computation of several quantities per mesh cell (e.g., slopes), the notion of "consistency" must be extended to this framework. A combination of consistency hypothesis with stability of the scheme leads to a suitable convergence theorem, generalizing the classical convergence theorem of Lax and Wendroff [52]. Finally, the limit functions are shown to be entropy solutions by using a notion of "Godunov compatibility", which serves as a substitute to the entropy condition. • The concept of "consistency" is explored in detail, for general high resolution schemes. • Consistency (and its order) of a finite difference scheme depends on the underlying space of approximation. • A generalization of the Lax-Wendroff convergence theorem is given. • The "entropy condition" is replaced by the concept of "Godunov compatibility". [ABSTRACT FROM AUTHOR]

Hypoxy induced angiogenesis processes can be described by coupling an integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the angiogenic factor. We propose high order positivity preserving schemes to approximate the marginal tip density by combining an asymptotic reduction with weighted essentially non oscillatory and strong stability preserving time discretization. We capture soliton-like solutions representing blood vessel formation and spread towards hypoxic regions. [ABSTRACT FROM AUTHOR]

In this work, we present a novel hybrid Discontinuous Galerkin scheme with hp-adaptivity capabilities for the compressible Euler equations. In smooth regions, an efficient and accurate discretization is achieved via local p-adaptation. At strong discontinuities and shocks, a finite volume scheme on an h-refined element-local subgrid gives robustness. Thus, we obtain a hp-adaptive scheme that exploits both the high convergence rate and efficiency of a p-adaptive high order scheme as well as the stable and accurate shock capturing abilities of a low order finite volume scheme, but avoids the inherent resolution loss through h-refinement. A single a priori indicator, based on the modal decay of the local polynomial solution representation, is used to distinguish between discontinuous and smooth regions and control the p-refinement. Our method is implemented as an extension to the open source software FLEXI. Hence, the efficient implementation of the method for high performance computers was an important criterion during the development. The efficiency of our adaptive scheme is demonstrated for a variety of test cases, where results are compared against non adaptive simulations. Our findings suggest that the proposed adaptive method produces comparable or even better results with significantly less computational costs. [ABSTRACT FROM AUTHOR]

• CFD simulation of heat convective-diffusive solidification of non-Newtonian fluid. • Fast-accurate prediction of heat transfer in phase change by SIMPLERnP/FVM scheme. • Ternary alloy solidification at high Rayleigh number with Herschel-Bulkley rheology. • Variations of power index and yield stress change leading heat transfer mechanism. • Temperature variation of liquid phase change fraction a key issue in ternary alloys. Numerical modeling of convective heat transfer employing non-Newtonian fluids is a key issue and represent an interesting challenge in many engineering applications. For this reason, this work develops an enhanced numerical model to predict fast and accurately the convective phase change for generalized non-Newtonian fluids. The strongly thermally coupled model is solved by an improved pressure-correction algorithm (SIMPLERnP) and the Finite Volume Method with accuracy of the third order for the transient terms and second order for the convective boundary conditions and the velocity gradients that allows to calculate the apparent viscosity. The study examines the natural heat convective solidification of the ternary Al-27 %Cu-5.25 %Si alloy with the Herschel-Bulkley rheology at a high Rayleigh number (>107). In addressing the main problem, four crucial aspects are discussed: the validation of the numerical scheme with numerical and experimental results; the modeling of the non-linear temperature variation of the liquid phase change fraction; the non-Newtonian effects of the power index (0.1 ≤ n ≤ 1.9) and yield stress (0≤ τ 0 ≤5 Pa) on the heat transfer mechanisms; and finally, the time of computation with the high-order numerical scheme and the enhanced pressure-correction algorithm. The main finding is that FVM/SIMPLERnP provides a significant reduction in the time of computation compared to reliable pressure-correction schemes (SIMPLER and IDEAL), being a robust, accurate and efficient scheme to solve complex natural heat convective phase change problems involve generalized non-Newtonian fluids at high-Rayleigh numbers. [ABSTRACT FROM AUTHOR]

New criteria for energy stability of multi-step, multi-stage, and mixed schemes are introduced in the context of evolution equations that arise as gradient flow with respect to a metric. These criteria are used to exhibit second and third order consistent, energy stable schemes, which are then demonstrated on several partial differential equations that arise as gradient flow with respect to the 2-Wasserstein metric. • Energy stability conditions for multi-step, multi-stage, and mixed schemes for gradient flow in arbitrary metric spaces. • New criteria are proposed based on embedding of finite graphs into Euclidean spaces. • Second order energy stable and third order energy bounded schemes are exhibited. [ABSTRACT FROM AUTHOR]

• A discretely (locally and globally) conservative high-order semi-discrete entropy-stable scheme has been presented. • The scheme conserves mass, momentum, energy, and satisfies the celebrated H-theorem at a discrete level. • To address low/high rarefaction flows, we couple the scheme with an iteration-free implicit-explicit time discretization. • The scheme yields stable solution for Riemann problems involving shocks even without using a shock-limiter. The kinetic equations govern the behavior of gaseous flows, compressibility, turbulence, reactions with internal energy exchange, that form the essence of numerous physical processes. Despite their wide applicability, their six-dimensional (seven including time) nature presents a huge computational challenge. With the advent of the modern high performance computing systems and few recent advances in numerical methods, it is now possible to numerically study the behavior of these systems. However, to understand the instability of rich molecular processes and find a structure in the apparent chaos, a scheme that is efficient in multi-dimensions, exhibits high parallel-efficiency, and foremost produces an entropy solution as per the theory of solutions of hyperbolic systems may prove useful. In this work, first, we construct an entropy stable flux for non-linear inhomogeneous (full) Boltzmann equation. Second, to ensure geometrical flexibility, we couple the scheme with a class of high order discontinuous Galerkin discretization (Jaiswal 2019 [41]) which satisfy summation-by-part (SBP) discretely. Third, utilizing SBP, we prove that the resulting semi-discrete scheme is locally and globally conservative in flat spaces; preserves the entropy-decay (H-theorem) property; efficient and simple; and therefore suitable for dealing with highly complex non-smooth flow problems. Fourth, we show that the fully-discrete kinetic scheme, utilizing an implicit-explicit time-discretization, in the limit of vanishing Knudsen number, becomes an entropy-stable explicit scheme applied to the Euler system. Fifth, we carry out a series of verification tests to illustrate the stability, accuracy, and conservation properties of the proposed method. These tests involve over 85 million degrees of freedom and over 50 billion operations per time-step. [ABSTRACT FROM AUTHOR]