29 research outputs found
Higher order finite difference schemes for the magnetic induction equations
We describe high order accurate and stable finite difference schemes for the
initial-boundary value problem associated with the magnetic induction
equations. These equations model the evolution of a magnetic field due to a
given velocity field. The finite difference schemes are based on Summation by
Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation
Term (SAT) technique for imposing boundary conditions. We present various
numerical experiments that demonstrate both the stability as well as high order
of accuracy of the schemes.Comment: 20 page
Strongly stable dual-pairing summation by parts finite difference schemes for the vector invariant nonlinear shallow water equations -- I: numerical scheme and validation
We present an energy/entropy stable and high order accurate finite difference
method for solving the linear/nonlinear shallow water equations (SWE) in vector
invariant form using the newly developed dual-pairing (DP) and
dispersion-relation preserving (DRP) summation by parts (SBP) finite difference
operators. We derive new well-posed boundary conditions for the SWE in one
space dimension, formulated in terms of fluxes and applicable to linear and
nonlinear problems. For nonlinear problems, entropy stability ensures the
boundedness of numerical solutions, however, it does not guarantee convergence.
Adequate amount of numerical dissipation is necessary to control high frequency
errors which could ruin numerical simulations. Using the dual-pairing SBP
framework, we derive high order accurate and nonlinear hyper-viscosity operator
which dissipates entropy and enstrophy. The hyper-viscosity operator
effectively tames oscillations from shocks and discontinuities, and eliminates
poisonous high frequency grid-scale errors. The numerical method is most
suitable for the simulations of sub-critical flows typical observed in
atmospheric and geostrophic flow problems. We prove a priori error estimates
for the semi-discrete approximations of both linear and nonlinear SWE. We
verify convergence, accuracy and well-balanced property via the method of
manufactured solutions (MMS) and canonical test problems such as the dam break,
lake at rest, and a two-dimensional rotating and merging vortex problem.Comment: 32 pages, 10 figures, comments are welcom
A Flux-Differencing Formula for Split-Form Summation By Parts Discretizations of Non-Conservative Systems: Applications to Subcell Limiting for magneto-hydrodynamics
In this paper, we show that diagonal-norm summation by parts (SBP)
discretizations of general non-conservative systems of hyperbolic balance laws
can be rewritten as a finite-volume-type formula, also known as
flux-differencing formula, if the non-conservative terms can be written as the
product of a local and a symmetric contribution. Furthermore, we show that the
existence of a flux-differencing formula enables the use of recent subcell
limiting strategies to improve the robustness of the high-order
discretizations.
To demonstrate the utility of the novel flux-differencing formula, we
construct hybrid schemes that combine high-order SBP methods (the discontinuous
Galerkin spectral element method and a high-order SBP finite difference method)
with a compatible low-order finite volume (FV) scheme at the subcell level. We
apply the hybrid schemes to solve challenging magnetohydrodynamics (MHD)
problems featuring strong shocks
Energy-conserving 3D elastic wave simulation with finite difference discretization on staggered grids with nonconforming interfaces
In this work, we describe an approach to stably simulate the 3D isotropic
elastic wave propagation using finite difference discretization on staggered
grids with nonconforming interfaces. Specifically, we consider simulation
domains composed of layers of uniform grids with different grid spacings,
separated by planar interfaces. This discretization setting is motivated by the
observation that wave speeds of earth media tend to increase with depth due to
sedimentation and consolidation processes. We demonstrate that the layer-wise
finite difference discretization approach has the potential to significantly
reduce the simulation cost, compared to its counterpart that uses holistically
uniform grids. Such discretizations are enabled by summation-by-parts finite
difference operators, which are standard finite difference operators with
special adaptations near boundaries or interfaces, and simultaneous
approximation terms, which are penalty terms appended to the discretized system
to weakly impose boundary or interface conditions. Combined with specially
designed interpolation operators, the discretized system is shown to preserve
the energy-conserving property of the continuous elastic wave equation, and a
fortiori ensure the stability of the simulation. Numerical examples are
presented to corroborate these analytical developments
Entropy Stable Summation-by-Parts Methods for Hyperbolic Conservation Laws on h/p Non-Conforming Meshes
In this work we present high-order primary conservative and entropy stable schemes for hyperbolic systems of conservation laws with geometric (h) and algebraic (p) non-conforming rectangular meshes. Throughout we rely on summation-by-parts (SBP) operators which discretely mimic the integration-by-parts rule to construct stable approximations. Thus, the discrete proofs of primary conservation and entropy stability can be done in a one-to-one fashion to the continuous analysis. Here, we consider different SBP operators based on finite difference as well as discontinuous Galerkin approaches. We derive non-conforming schemes by extending ideas of high-order primary conservative and entropy stable SBP methods on conforming meshes. Here, special attention is given to the coupling between non-conforming elements. The coupling is instructed to entropy stable projection operators. However, these projection operators suffer from a suboptimal degree. Therefore, we develop degree preserving SBP operators where the norm matrix has a higher degree compared to classical SBP operators. With these operators it is possible to construct entropy stable projection operators which have the same degree as the SBP differentiationmatrix. Typically, high-order primary conservative and entropy stable schemes are semi-discrete methods with a discretized spatial domain and assuming continuity in time. Therefore, temporal errors are introduced when integrating the conservation laws in time with standard methods, e.g. Runge-Kutta schemes, for which the entropy can have an unpredictable temporal behaviour. Thus, we extend high-order primary conservative and entropy stable semi-discrete methods to fully-discrete schemes on conforming and non-conforming meshes. This results in an implicit space-time method. We introduce a simple mesh generation strategy to obtain quadrilateral meshes surrounding two dimensional complex geometries. Finally, with the generated meshes we simulate a flow around a NACA0012 airfoil to demonstrate the benefits of considering non-conforming elements for a practical simulation