This paper introduces the method of anisotropic diffusion oscillation reduction (ADOR) for shock wave computations. The connection is made between digital image processing,in particular, image edge detection, and numerical shock capturing. Indeed, numerical shock capturing can be formulated on the lines of iterative digital edge detection. Various anisotropic diffusion and super diffusion operators originated from image edge detection are proposed for the treatment of hyperbolic conservation laws and near-hyperbolic hydrodynamic equations of change. The similarity between anisotropic diffusion and artificial viscosity is discussed. Physical origins and mathematical properties of the artificial viscosity is analyzed from the kinetic theory point of view. A form of pressure tensor is derived from the first principles of the quantum mechanics. Quantum kinetic theory is utilized to arrive at macroscopic transport equations from the microscopic theory. Macroscopic symmetry is used to simplify pressure tensor expressions. The latter provides a basis for the design of artificial viscosity. The ADOR approach is validated by using (inviscid) Burgers' equation in one and two spatial dimensions, the incompressible Navier-Stokes equation and the Euler equation. A discrete singular convolution (DSC) algorithm is utilized for the spatial discretization.
A main disadvantage of many high-order methods for hyperbolic conservation laws lies in the famous Gibbs-Wilbraham phenomenon, once discontinuities appear in the solution. Due to the Gibbs-Wilbraham phenomenon, the numerical approximation will be polluted by spurious oscillations, which produce unphysical numerical solutions and might finally blow up the computation. In this work, we propose a new shock capturing procedure to stabilise high-order spectral element approximations. The procedure consists of going over from the original (polluted) approximation to a convex combination of the origi…
A new procedure to capture the shocks has been proposed and is demonstrated for the solutions of two-dimensional Euler equations using discontinuous Galerkin method and overset grids. A discontinuous Galerkin solver using a coarse grid provides the troubled cell data that is used to determine the location of the shock. An overset grid aligned to the shock is constructed based on this information. The solver is run again with the coarse grid solution as the initial condition with limiting occurring only in the overset grid to give a solution with the shock aligned to the grid line. Results for …
This article considers a new discretization scheme for conservation laws. The discretization setting is based on a discontinuous Galerkin scheme in combination with an approximation space that contains high-order polynomial modes as well as piece-wise constant modes on a sub-grid. The high-order modes can continuously be suppressed with a penalty function that is based on a sensor which is intertwined with the approximation space. Numerical tests finally illustrate the performance of this scheme.
We present a novel structure-preserving numerical scheme for discontinuous finite element approximations of nonlinear hyperbolic systems. The method can be understood as a generalization of the Lax-Friedrichs flux to a high-order staggered grid and does not depend on any tunable parameters. Under a presented set of conditions, we show that the method is conservative and invariant domain preserving. Numerical experiments on the Euler equations show the ability of the scheme to resolve discontinuities without introducing excessive spurious oscillations or dissipation.
This note introduces a simple metric for benchmarking shock-capturing schemes. This metric is especially focused on the shock-capturing overshoots, which may undermine the robustness of numerical simulations, as well as the reliability of numerical results. The idea is to numerically solve the model linear advection equation with an initial condition of a square wave characterized with different wavenumbers. After one step of temporal evolution, the exact numerical overshoot error can be easily determined and shown as a function of the CFL number and the reduced wavenumber. With the overshoot …
We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. (B.W. Wissink, G.B. Jacobs, J.K. Ryan, W.S. Don, and E.T.A. van der Weide. Shock regularization with smoothness-increasing accuracy-conserving Dirac-delta polynomial kernels. Journal of Scientific Computing, 77:579--596, 2018). In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well kn…
In this paper, a new strategy for a sub-element based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low to high order discretizations on this set of data, including a first order finite volume scheme up to the full order DG scheme. The different DG discretizations are then blended according to sub-element troubled cell indicators, resulting in a final discretization that adaptively blends from low to high order within a single DG element. The goal is to retain as much high …
h-Adaptive FV Subcell Shock-Capturing for DGSEM on Heterogeneous Curvilinear Meshes
Institute of Aerodynamics and Gas Dynamics, University of Stuttgart（斯图加特大学空气动力学与气体动力学研究所）；Institute of Applied Analysis and Numerical Simulation, University of Stuttgart（斯图加特大学应用分析与数值模拟研究所）
In this work, we introduce a novel approach to formulating an artificial viscosity for shock capturing in nonlinear hyperbolic systems by utilizing the property that the solutions of hyperbolic conservation laws are not reversible in time in the vicinity of shocks. The proposed approach does not require any additional governing equations or a priori knowledge of the hyperbolic system in question, is independent of the mesh and approximation order, and requires the use of only one tunable parameter. The primary novelty is that the resulting artificial viscosity is unique for each component of t…
In this work, a framework to construct arbitrarily high-order low-dissipation shock-capturing schemes with flexible and controllable nonlinear dissipation for convection-dominated problems is proposed. While a set of candidate stencils of incremental width is constructed, each one is indicated as smooth or nonsmooth by the ENO-like stencil selection procedure proposed in the targeted essentially non-oscillatory (TENO) scheme [Fu et al., Journal of Computational Physics 305 (2016): 333-359]. Rather than being discarded directly as with TENO schemes, the nonsmooth candidates are filtered by an e…
In this paper we enhance the well-known fifth order WENO shock-capturing scheme by using deep learning techniques. This fine-tuning of an existing algorithm is implemented by training a rather small neural network to modify the smoothness indicators of the WENO scheme in order to improve the numerical results especially at discontinuities. In our approach no further post-processing is needed to ensure the consistency of the method, which simplifies the method and increases the effect of the neural network. Moreover, the convergence of the resulting scheme can be theoretically proven. We demons…
We present a stable spectral vanishing viscosity for discontinuous Galerkin schemes, with applications to turbulent and supersonic flows. The idea behind the SVV is to spatially filter the dissipative fluxes, such that it concentrates in higher wavenumbers, where the flow is typically under-resolved, leaving low wavenumbers dissipation-free. Moreover, we derive a stable approximation of the Guermond-Popov fluxes with the Bassi-Rebay 1 scheme, used to introduce density regularization in shock capturing simulations. This filtering uses a Cholesky decomposition of the fluxes that ensures the entr…
Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes
Institute of Mathematics, Johannes Gutenberg University（约翰内斯·古腾堡大学数学研究所）；High-Performance Scientific Computing, Centre for Advanced Analytics and Predictive Sciences, University of Augsburg（奥格斯堡大学高级分析与预测科学中心高性能科学计算）；Mathematical Institute, University of Cologne（科隆大学数学研究所）
In this paper, we present a novel hybrid nonlinear explicit-compact scheme for shock-capturing based on a boundary variation diminishing (BVD) reconstruction. In our approach, we combine a non-dissipative sixth-order central compact interpolation and a fifth-order monotonicity preserving scheme (MP5) through the BVD algorithm. For a smooth solution, the BVD reconstruction chooses the highest order possible interpolation, which is central, i.e. non-dissipative in the current approach and for the discontinuities, the algorithm selects the monotone scheme. This method provides an alternative to t…
We present a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method uses continuous Lagrange polynomials in space and explicit Runge-Kutta schemes in time. The shock-capturing term is based on the residual of MHD which tracks the shock and discontinuity positions, and adds a sufficient amount of viscosity to stabilize them. The method is tested up to third order polynomial spaces and an expected fourth-order convergence rate is obtained for smooth problems. Several discontinuous benchmarks such as Orszag-Tang, MHD rotor, Brio-Wu prob…
A categorical embedding discontinuity-capturing shallow neural network for anisotropic elliptic interface problems
Department of Mathematics, National Central University（数学系，国立中央大学）；Department of Applied Mathematics, National Yang Ming Chiao Tung University（应用数学系，国立阳明交通大学）；Department of Applied Mathematics, National University of Kaohsiung（应用数学系，国立高雄大学）
The second paper of this series presents two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element (DGSEM) discretizations of the compressible magneto-hydrodynamics (MHD) equations. Specifically, we use the resistive GLM-MHD equations, which include a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM). For the continuous entropy analysis to hold, and due to the divergence-free constraint on the magnetic field, the GLM-MHD system requires the use of non-conservative terms, which need special treatment. Hennemann et al. [DOI…
The stable and accurate approximation of discontinuities such as shocks on a finite computational mesh is a challenging task. Detection of shocks or strong discontinuities in the flow solution is typically achieved through a priori troubled cell indicators, which guide the subsequent action of an appropriate shock capturing mechanism. Arriving at a stable and accurate solution often requires empirically based parameter tuning and adjustments of the indicator settings to the discretization and solution at hand. In this work, we propose to separate the task of shock detection and shock capturing…
In recent years, machine learning has been used to create data-driven solutions to problems for which an algorithmic solution is intractable, as well as fine-tuning existing algorithms. This research applies machine learning to the development of an improved finite-volume method for simulating PDEs with discontinuous solutions. Shock capturing methods make use of nonlinear switching functions that are not guaranteed to be optimal. Because data can be used to learn nonlinear relationships, we train a neural network to improve the results of a fifth-order WENO method. We post-process the outputs…
The nonlinear convection terms in the governing equations of compressible fluid flows are hyperbolic in nature and are nontrivial for modelling and numerical simulation. Many numerical methods have been developed in the last few decades for this purpose and are typically based on Riemann solvers, which are strongly dependent on the underlying eigen-structure of the governing equations. Objective of the present work is to develop simple algorithms which are not dependent on the eigen-structure and yet can tackle easily the hyperbolic parts. Central schemes with smart diffusion mechanisms are ap…
We propose a change of variable approach and discontinuity capturing methods to ensure physical constraints for advection-reaction equations discretized by the finite element method. This change of variable confines the concentration below an upper bound in a very natural way. For the non-negativity constraint, we propose to use a discontinuity capturing method defined on the reference element that is combined with an anisotropic crosswind-dissipation operator. This discontinuity capturing cannot completely eliminate negative values but effectively minimizes their occurrence. The proposed meth…
Scientists and engineers often create accurate, trustworthy, computational simulation schemes - but all too often these are too computationally expensive to execute over the time or spatial domain of interest. The equation-free approach is to marry such trusted simulations to a framework for numerical macroscale reduction - the patch dynamics scheme. This article extends the patch scheme to scenarios in which the trusted simulation resolves abrupt state changes on the microscale that appear as shocks on the macroscale. Accurate simulation for problems in these scenarios requires extending the …
