26 research outputs found
A Hybrid High-Order method for nonlinear elasticity
In this work we propose and analyze a novel Hybrid High-Order discretization
of a class of (linear and) nonlinear elasticity models in the small deformation
regime which are of common use in solid mechanics. The proposed method is valid
in two and three space dimensions, it supports general meshes including
polyhedral elements and nonmatching interfaces, enables arbitrary approximation
order, and the resolution cost can be reduced by statically condensing a large
subset of the unknowns for linearized versions of the problem. Additionally,
the method satisfies a local principle of virtual work inside each mesh
element, with interface tractions that obey the law of action and reaction. A
complete analysis covering very general stress-strain laws is carried out, and
optimal error estimates are proved. Extensive numerical validation on model
test problems is also provided on two types of nonlinear models.Comment: 29 pages, 7 figures, 4 table
Convergence of summation-by-parts finite difference methods for the wave equation
In this paper, we consider finite difference approximations of the second
order wave equation. We use finite difference operators satisfying the
summation-by-parts property to discretize the equation in space. Boundary
conditions and grid interface conditions are imposed by the
simultaneous-approximation-term technique. Typically, the truncation error is
larger at the grid points near a boundary or grid interface than that in the
interior. Normal mode analysis can be used to analyze how the large truncation
error affects the convergence rate of the underlying stable numerical scheme.
If the semi-discretized equation satisfies a determinant condition, two orders
are gained from the large truncation error. However, many interesting second
order equations do not satisfy the determinant condition. We then carefully
analyze the solution of the boundary system to derive a sharp estimate for the
error in the solution and acquire the gain in convergence rate. The result
shows that stability does not automatically yield a gain of two orders in
convergence rate. The accuracy analysis is verified by numerical experiments.Comment: In version 2, we have added a new section on the convergence analysis
of the Neumann problem, and have improved formulations in many place
High-order accurate well-balanced energy stable adaptive moving mesh finite difference schemes for the shallow water equations with non-flat bottom topography
This paper proposes high-order accurate well-balanced (WB) energy stable (ES)
adaptive moving mesh finite difference schemes for the shallow water equations
(SWEs) with non-flat bottom topography. To enable the construction of the ES
schemes on moving meshes, a reformulation of the SWEs is introduced, with the
bottom topography as an additional conservative variable that evolves in time.
The corresponding energy inequality is derived based on a modified energy
function, then the reformulated SWEs and energy inequality are transformed into
curvilinear coordinates. A two-point energy conservative (EC) flux is
constructed, and high-order EC schemes based on such a flux are proved to be WB
that they preserve the lake at rest. Then high-order ES schemes are derived by
adding suitable dissipation terms to the EC schemes, which are newly designed
to maintain the WB and ES properties simultaneously. The adaptive moving mesh
strategy is performed by iteratively solving the Euler-Lagrangian equations of
a mesh adaptation functional. The fully-discrete schemes are obtained by using
the explicit strong-stability preserving third-order Runge-Kutta method.
Several numerical tests are conducted to validate the accuracy, WB and ES
properties, shock-capturing ability, and high efficiency of the schemes.Comment: 40 pages, 16 figure
An Improved High Order Finite Difference Method for Non-conforming Grid Interfaces for the Wave Equation
This paper presents an extension of a recently developed high order finite difference method for the wave equation on a grid with non-conforming interfaces. The stability proof of the existing methods relies on the interpolation operators being norm-contracting, which is satisfied by the second and fourth order operators, but not by the sixth order operator. We construct new penalty terms to impose interface conditions such that the stability proof does not require the norm-contracting condition. As a consequence, the sixth order accurate scheme is also provably stable. Numerical experiments demonstrate the improved stability and accuracy property
An Improved High Order Finite Difference Method for Non-conforming Grid Interfaces for the Wave Equation
This paper presents an extension of a recently developed high order finite difference method for the wave equation on a grid with non-conforming interfaces. The stability proof of the existing methods relies on the interpolation operators being norm-contracting, which is satisfied by the second and fourth order operators, but not by the sixth order operator. We construct new penalty terms to impose interface conditions such that the stability proof does not require the norm-contracting condition. As a consequence, the sixth order accurate scheme is also provably stable. Numerical experiments demonstrate the improved stability and accuracy property
High-order accurate well-balanced energy stable finite difference schemes for multi-layer shallow water equations on fixed and adaptive moving meshes
This paper develops high-order well-balanced (WB) energy stable (ES) finite
difference schemes for multi-layer (the number of layers )
shallow water equations (SWEs) on both fixed and adaptive moving meshes,
extending our previous works [20,51]. To obtain an energy inequality, the
convexity of an energy function for an arbitrary is proved by finding
recurrence relations of the leading principal minors or the quadratic forms of
the Hessian matrix of the energy function with respect to the conservative
variables, which is more involved than the single-layer case due to the
coupling between the layers in the energy function. An important ingredient in
developing high-order semi-discrete ES schemes is the construction of a
two-point energy conservative (EC) numerical flux. In pursuit of the WB
property, a sufficient condition for such EC fluxes is given with compatible
discretizations of the source terms similar to the single-layer case. It can be
decoupled into identities individually for each layer, making it convenient
to construct a two-point EC flux for the multi-layer system. To suppress
possible oscillations near discontinuities, WENO-based dissipation terms are
added to the high-order WB EC fluxes, which gives semi-discrete high-order WB
ES schemes. Fully-discrete schemes are obtained by employing high-order
explicit SSP-RK methods and proved to preserve the lake at rest. The schemes
are further extended to moving meshes based on a modified energy function for a
reformulated system, relying on the techniques proposed in [51]. Numerical
experiments are conducted for some two- and three-layer cases to validate the
high-order accuracy, WB and ES properties, and high efficiency of the schemes,
with a suitable amount of dissipation chosen by estimating the maximal wave
speed due to the lack of an analytical expression for the eigenstructure of the
multi-layer system.Comment: 54 pages, 19 figure