51 research outputs found
GQL-Based Bound-Preserving and Locally Divergence-Free Central Discontinuous Galerkin Schemes for Relativistic Magnetohydrodynamics
This paper develops novel and robust central discontinuous Galerkin (CDG)
schemes of arbitrarily high-order accuracy for special relativistic
magnetohydrodynamics (RMHD) with a general equation of state (EOS). These
schemes are provably bound-preserving (BP), i.e., consistently preserve the
upper bound for subluminal fluid velocity and the positivity of density and
pressure, while also (locally) maintaining the divergence-free (DF) constraint
for the magnetic field. For 1D RMHD, the standard CDG method is exactly DF, and
its BP property is proven under a condition achievable by BP limiter. For 2D
RMHD, we design provably BP and locally DF CDG schemes based on the suitable
discretization of a modified RMHD system. A key novelty in our schemes is the
discretization of additional source terms in the modified RMHD equations, so as
to precisely counteract the influence of divergence errors on the BP property
across overlapping meshes. We provide rigorous proofs of the BP property for
our CDG schemes and first establish the theoretical connection between BP and
discrete DF properties on overlapping meshes for RMHD. Owing to the absence of
explicit expressions for primitive variables in terms of conserved variables,
the constraints of physical bounds are strongly nonlinear, making the BP proofs
highly nontrivial. We overcome these challenges through technical estimates
within the geometric quasilinearization (GQL) framework, which converts the
nonlinear constraints into linear ones. Furthermore, we introduce a new 2D cell
average decomposition on overlapping meshes, which relaxes the theoretical BP
CFL constraint and reduces the number of internal nodes, thereby enhancing the
efficiency of the 2D BP CDG method. We implement the proposed CDG schemes for
extensive RMHD problems with various EOSs, demonstrating their robustness and
effectiveness in challenging scenarios.Comment: 47 pages, 14 figure
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Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)
This small collaborative workshop brought together
experts from the Sino-German project working in the field of advanced numerical methods for
hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the
convergence of numerical methods and proper solution concepts were addressed as well
Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations
Discontinuous Galerkin (DG) methods have a long history in computational
physics and engineering to approximate solutions of partial differential
equations due to their high-order accuracy and geometric flexibility. However,
DG is not perfect and there remain some issues. Concerning robustness, DG has
undergone an extensive transformation over the past seven years into its modern
form that provides statements on solution boundedness for linear and nonlinear
problems.
This chapter takes a constructive approach to introduce a modern incarnation
of the DG spectral element method for the compressible Navier-Stokes equations
in a three-dimensional curvilinear context. The groundwork of the numerical
scheme comes from classic principles of spectral methods including polynomial
approximations and Gauss-type quadratures. We identify aliasing as one
underlying cause of the robustness issues for classical DG spectral methods.
Removing said aliasing errors requires a particular differentiation matrix and
careful discretization of the advective flux terms in the governing equations.Comment: 85 pages, 2 figures, book chapte
A Physical-Constraint-Preserving Finite Volume WENO Method for Special Relativistic Hydrodynamics on Unstructured Meshes
This paper presents a highly robust third-order accurate finite volume
weighted essentially non-oscillatory (WENO) method for special relativistic
hydrodynamics on unstructured triangular meshes. We rigorously prove that the
proposed method is physical-constraint-preserving (PCP), namely, always
preserves the positivity of the pressure and the rest-mass density as well as
the subluminal constraint on the fluid velocity. The method is built on a
highly efficient compact WENO reconstruction on unstructured meshes, a simple
PCP limiter, the provably PCP property of the Harten--Lax--van Leer flux, and
third-order strong-stability-preserving time discretization. Due to the
relativistic effects, the primitive variables (namely, the rest-mass density,
velocity, and pressure) are highly nonlinear implicit functions in terms of the
conservative variables, making the design and analysis of our method
nontrivial. To address the difficulties arising from the strong nonlinearity,
we adopt a novel quasilinear technique for the theoretical proof of the PCP
property. Three provable convergence-guaranteed iterative algorithms are also
introduced for the robust recovery of primitive quantities from admissible
conservative variables. We also propose a slight modification to an existing
WENO reconstruction to ensure the scaling invariance of the nonlinear weights
and thus to accommodate the homogeneity of the evolution operator, leading to
the advantages of the modified WENO reconstruction in resolving multi-scale
wave structures. Extensive numerical examples are presented to demonstrate the
robustness, expected accuracy, and high resolution of the proposed method.Comment: 56 pages, 18 figure
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