7,722 research outputs found
On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations
The aim of this paper is to extend the global error estimation and control
addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value
problems to finite difference solutions of semilinear parabolic partial
differential equations. The approach presented there is combined with an
estimation of the PDE spatial truncation error by Richardson extrapolation to
estimate the overall error in the computed solution. Approximations of the
error transport equations for spatial and temporal global errors are derived by
using asymptotic estimates that neglect higher order error terms for
sufficiently small step sizes in space and time. Asymptotic control in a
discrete -norm is achieved through tolerance proportionality and uniform
or adaptive mesh refinement. Numerical examples are used to illustrate the
reliability of the estimation and control strategies
Parabolic and Hyperbolic Contours for Computing the Bromwich Integral
Some of the most effective methods for the numerical inversion of the Laplace transform are based on the approximation of the Bromwich contour integral. The accuracy of these methods often hinges on a good choice of contour, and several such contours have been proposed in the literature. Here we analyze two recently proposed contours, namely a parabola and a hyperbola. Using a representative model problem, we determine estimates for the optimal parameters that define these contours. An application to a fractional diffusion equation is presented.\ud
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JACW was supported by the National Research Foundation in South Africa under grant FA200503230001
CASTRO: A New Compressible Astrophysical Solver. II. Gray Radiation Hydrodynamics
We describe the development of a flux-limited gray radiation solver for the
compressible astrophysics code, CASTRO. CASTRO uses an Eulerian grid with
block-structured adaptive mesh refinement based on a nested hierarchy of
logically-rectangular variable-sized grids with simultaneous refinement in both
space and time. The gray radiation solver is based on a mixed-frame formulation
of radiation hydrodynamics. In our approach, the system is split into two
parts, one part that couples the radiation and fluid in a hyperbolic subsystem,
and another parabolic part that evolves radiation diffusion and source-sink
terms. The hyperbolic subsystem is solved explicitly with a high-order Godunov
scheme, whereas the parabolic part is solved implicitly with a first-order
backward Euler method.Comment: accepted for publication in ApJS, high-resolution version available
at https://ccse.lbl.gov/Publications/wqzhang/castro2.pd
A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: Triangular grids
A novel wetting and drying treatment for second-order Runge-Kutta
discontinuous Galerkin (RKDG2) methods solving the non-linear shallow water
equations is proposed. It is developed for general conforming two-dimensional
triangular meshes and utilizes a slope limiting strategy to accurately model
inundation. The method features a non-destructive limiter, which concurrently
meets the requirements for linear stability and wetting and drying. It further
combines existing approaches for positivity preservation and well-balancing
with an innovative velocity-based limiting of the momentum. This limiting
controls spurious velocities in the vicinity of the wet/dry interface. It leads
to a computationally stable and robust scheme -- even on unstructured grids --
and allows for large time steps in combination with explicit time integrators.
The scheme comprises only one free parameter, to which it is not sensitive in
terms of stability. A number of numerical test cases, ranging from analytical
tests to near-realistic laboratory benchmarks, demonstrate the performance of
the method for inundation applications. In particular, super-linear
convergence, mass-conservation, well-balancedness, and stability are verified
Efficient hierarchical approximation of high-dimensional option pricing problems
A major challenge in computational finance is the pricing of options that depend on a large number of risk factors. Prominent examples are basket or index options where dozens or even hundreds of stocks constitute the underlying asset and determine the dimensionality of the corresponding degenerate parabolic equation. The objective of this article is to show how an efficient discretisation can be achieved by hierarchical approximation as well as asymptotic expansions of the underlying continuous problem. The relation to a number of state-of-the-art methods is highlighted
Multi-Adaptive Time-Integration
Time integration of ODEs or time-dependent PDEs with required resolution of
the fastest time scales of the system, can be very costly if the system
exhibits multiple time scales of different magnitudes. If the different time
scales are localised to different components, corresponding to localisation in
space for a PDE, efficient time integration thus requires that we use different
time steps for different components.
We present an overview of the multi-adaptive Galerkin methods mcG(q) and
mdG(q) recently introduced in a series of papers by the author. In these
methods, the time step sequence is selected individually and adaptively for
each component, based on an a posteriori error estimate of the global error.
The multi-adaptive methods require the solution of large systems of nonlinear
algebraic equations which are solved using explicit-type iterative solvers
(fixed point iteration). If the system is stiff, these iterations may fail to
converge, corresponding to the well-known fact that standard explicit methods
are inefficient for stiff systems. To resolve this problem, we present an
adaptive strategy for explicit time integration of stiff ODEs, in which the
explicit method is adaptively stabilised by a small number of small,
stabilising time steps
CASTRO: A New Compressible Astrophysical Solver. III. Multigroup Radiation Hydrodynamics
We present a formulation for multigroup radiation hydrodynamics that is
correct to order using the comoving-frame approach and the
flux-limited diffusion approximation. We describe a numerical algorithm for
solving the system, implemented in the compressible astrophysics code, CASTRO.
CASTRO uses an Eulerian grid with block-structured adaptive mesh refinement
based on a nested hierarchy of logically-rectangular variable-sized grids with
simultaneous refinement in both space and time. In our multigroup radiation
solver, the system is split into three parts, one part that couples the
radiation and fluid in a hyperbolic subsystem, another part that advects the
radiation in frequency space, and a parabolic part that evolves radiation
diffusion and source-sink terms. The hyperbolic subsystem and the frequency
space advection are solved explicitly with high-order Godunov schemes, whereas
the parabolic part is solved implicitly with a first-order backward Euler
method. Our multigroup radiation solver works for both neutrino and photon
radiation.Comment: accepted by ApJS, 27 pages, 20 figures, high-resolution version
available at https://ccse.lbl.gov/Publications/wqzhang/castro3.pd
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