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Register a new userResidual viscosity stabilized RBF-FD methods for solving nonlinear conservation laws
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Added by
Igor Tominec
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We formulate an oversampled radial basis function generated finite difference (RBF-FD) method to solve time-dependent nonlinear conservation laws. The analytic solutions of these problems are known to be discontinuous, which leads to occurrence of non-physical oscillations (Gibbs phenomenon) that pollute the numerical solutions and can make them unstable. We address these difficulties using a residual based artificial viscosity stabilization, where the residual of the conservation law indicates the approximate location of the shocks. The location is then used to locally apply an upwind viscosity term, which stabilizes the Gibbs phenomenon and does not smear the solution away from the shocks. The proposed method is numerically tested and proves to be robust and accurate when solving scalar conservation laws and systems of conservation laws, such as compressible Euler equations.
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It is well understood that boundary conditions (BCs) may cause global radial basis function (RBF) methods to become unstable for hyperbolic conservation laws (CLs). Here we investigate this phenomenon and identify the strong enforcement of BCs as the mechanism triggering such stability issues. Based on this observation we propose a technique to weakly enforce BCs in RBF methods. In the case of hyperbolic CLs, this is achieved by carefully building RBF methods from the weak form of the CL, rather than the typically enforced strong form. Furthermore, we demonstrate that global RBF methods may violate conservation, yielding physically unreasonable solutions when the approximation does not take into account these considerations. Numerical experiments validate our theoretical results.
Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to the computational domain $Omega$. One of the requirements leading to approximation robustness is to place the interpolation points with a locally uniform distance around the boundary of $Omega$. However generating interpolation points with such properties is a cumbersome problem. Instead, the interpolation points can be extended over the boundary and as such completely decoupled from the shape of $Omega$. In this paper we present a modification to the least-squares RBF-FD method which allows the interpolation points to be placed in a box that encapsulates $Omega$. This way, the node placement over a complex domain in 2D and 3D is greatly simplified. Numerical experiments on solving an elliptic model PDE over complex 2D geometries show that our approach is robust. Furthermore it performs better in terms of the approximation error and the runtime vs. error compared with the classic RBF-FD methods. It is also possible to use our approach in 3D, which we indicate by providing convergence results of a solution over a thoracic diaphragm.
We propose an Exponential DG approach for numerically solving partial differential equations (PDEs). The idea is to decompose the governing PDE operators into linear (fast dynamics extracted by linearization) and nonlinear (the remaining after removing the former) parts, on which we apply the discontinuous Galerkin (DG) spatial discretization. The resulting semi-discrete system is then integrated using exponential time-integrators: exact for the former and approximate for the latter. By construction, our approach i) is stable with a large Courant number (Cr > 1); ii) supports high-order solutions both in time and space; iii) is computationally favorable compared to IMEX DG methods with no preconditioner; iv) requires comparable computational time compared to explicit RKDG methods, while having time stepsizes orders magnitude larger than maximal stable time stepsizes for explicit RKDG methods; v) is scalable in a modern massively parallel computing architecture by exploiting Krylov-subspace matrix-free exponential time integrators and compact communication stencil of DG methods. Various numerical results for both Burgers and Euler equations are presented to showcase these expected properties. For Burgers equation, we present detailed stability and convergence analyses for the exponential Euler DG scheme.
In this paper, we propose a hybrid finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme for solving one and two dimensional hyperbolic conservation laws. The zeroth-order and the first-order moments are used in the spatial reconstruction, with total variation diminishing Runge-Kutta time discretization. The main idea of the hybrid HWENO scheme is that we first use a shock-detection technique to identify the troubled cell, then, if the cell is identified as a troubled cell, we would modify the first order moment in the troubled cell and employ HWENO reconstruction in spatial discretization; otherwise, we directly use high order linear reconstruction. Unlike other HWENO schemes, we borrow the thought of limiter for discontinuous Galerkin (DG) method to control the spurious oscillations, after this procedure, the scheme would avoid the oscillations by using HWENO reconstruction nearby discontinuities and have higher efficiency for using linear approximation straightforwardly in the smooth regions. In addition, the hybrid HWENO scheme still keeps the compactness. A collection of benchmark numerical tests for one and two dimensional cases are performed to demonstrate the numerical accuracy, high resolution and robustness of the proposed scheme.
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 original approximation and its Bernstein reconstruction, yielding a stabilised approximation. The coefficient in the convex combination, and therefore the procedure, is steered by a discontinuity sensor and is only activated in troubled elements. Building up on classical Bernstein operators, we are thus able to prove that the resulting Bernstein procedure is total variation diminishing and preserves monotone (shock) profiles. Further, the procedure can be modified to not just preserve but also to enforce certain bounds for the solution, such as positivity. In contrast to other shock capturing methods, e.g. artificial viscosity methods, the new procedure does not reduce the time step or CFL condition and can be easily and efficiently implemented into any existing code. Numerical tests demonstrate that the proposed shock-capturing procedure is able to stabilise and enhance spectral element approximations in the presence of shocks.
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