10 research outputs found
A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions
A new parallel, computationally efficient immersed boundary method for
solving three-dimensional, viscous, incompressible flows on unbounded domains
is presented. Immersed surfaces with prescribed motions are generated using the
interpolation and regularization operators obtained from the discrete delta
function approach of the original (Peskin's) immersed boundary method. Unlike
Peskin's method, boundary forces are regarded as Lagrange multipliers that are
used to satisfy the no-slip condition. The incompressible Navier-Stokes
equations are discretized on an unbounded staggered Cartesian grid and are
solved in a finite number of operations using lattice Green's function
techniques. These techniques are used to automatically enforce the natural
free-space boundary conditions and to implement a novel block-wise adaptive
grid that significantly reduces the run-time cost of solutions by limiting
operations to grid cells in the immediate vicinity and near-wake region of the
immersed surface. These techniques also enable the construction of practical
discrete viscous integrating factors that are used in combination with
specialized half-explicit Runge-Kutta schemes to accurately and efficiently
solve the differential algebraic equations describing the discrete momentum
equation, incompressibility constraint, and no-slip constraint. Linear systems
of equations resulting from the time integration scheme are efficiently solved
using an approximation-free nested projection technique. The algebraic
properties of the discrete operators are used to reduce projection steps to
simple discrete elliptic problems, e.g. discrete Poisson problems, that are
compatible with recent parallel fast multipole methods for difference
equations. Numerical experiments on low-aspect-ratio flat plates and spheres at
Reynolds numbers up to 3,700 are used to verify the accuracy and physical
fidelity of the formulation.Comment: 32 pages, 9 figures; preprint submitted to Journal of Computational
Physic
Fault tolerance in an inner-outer solver: a GVR-enabled case study
Abstract. Resilience is a major challenge for large-scale systems. It is particularly important for iterative linear solvers, since they take much of the time of many scientific applications. We show that single bit flip errors in the Flexible GMRES iterative linear solver can lead to high computational overhead or even failure to converge to the right answer. Informed by these results, we design and evaluate several strategies for fault tolerance in both inner and outer solvers appropriate across a range of error rates. We implement them, extending Trilinos' solver library with the Global View Resilience (GVR) programming model, which provides multi-stream snapshots, multi-version data structures with portable and rich error checking/recovery. Experimental results validate correct execution with low performance overhead under varied error conditions
Efficient, positive, and energy stable schemes for multi-D Poisson-Nernst-Planck systems
In this paper, we design, analyze, and numerically validate positive and
energy-dissipating schemes for solving the time-dependent multi-dimensional
system of Poisson-Nernst-Planck (PNP) equations, which has found much use in
the modeling of biological membrane channels and semiconductor devices. The
semi-implicit time discretization based on a reformulation of the system gives
a well-posed elliptic system, which is shown to preserve solution positivity
for arbitrary time steps. The first order (in time) fully-discrete scheme is
shown to preserve solution positivity and mass conservation unconditionally,
and energy dissipation with only a mild time step restriction. The
scheme is also shown to preserve the steady-state. For the fully second order
(in both time and space) scheme with large time steps, solution positivity is
restored by a local scaling limiter, which is shown to maintain the spatial
accuracy. These schemes are easy to implement. Several three-dimensional
numerical examples verify our theoretical findings and demonstrate the
accuracy, efficiency, and robustness of the proposed schemes, as well as the
fast approach to steady states.Comment: 32 pages, 3 tables, 4 figure
A Family of Iteration Functions for General Linear Systems
We develop novel theory and algorithms for computing approximate solution to
, or to , where is an real matrix of
arbitrary rank. First, we describe the {\it Triangle Algorithm} (TA), where
given an ellipsoid , in each
iteration it either computes successively improving approximation , or proves . We then extend TA for
computing an approximate solution or minimum-norm solution. Next, we develop a
dynamic version of TA, the {\it Centering Triangle Algorithm} (CTA), generating
residuals via iterations of the simple formula,
, where when is symmetric PSD, otherwise
but need not be computed explicitly. More generally, CTA extends to a
family of iteration function, , satisfying: On the one
hand, given and , where with arbitrary, for all , and
converges to zero. Algorithmically, if is invertible with
condition number , in
iterations . If is singular with
the ratio of its largest to smallest positive eigenvalues, in iterations either
or . If is the number of nonzero
entries of , each iteration take operations. On the other hand,
given , suppose its minimal polynomial with respect to has
degree . Then is solvable if and only if . Moreover,
exclusively is solvable, if and only if but
. Additionally, is computable in
operations.Comment: 59 pages, 4 figure
A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions
A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier–Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge–Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recent parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation
Positive and energy stable schemes for Poisson-Nernst-Planck equations and related models
In this thesis, we design, analyze, and numerically validate positive and energy-
dissipating schemes for solving Poisson-Nernst-Planck (PNP) equations and Fokker-Planck (FP) equations with interaction potentials. These equations play an important role in modeling the dynamics of charged particles in semiconductors and biological ion channels, as well as in other applications. These model equations are nonlinear/nonlocal gradient flows in density space, and their explicit solutions are rarely available; however, solutions to such problems feature intrinsic properties such as (i) solution positivity, (ii) mass conservation, and (iii) energy dissipation. These physically relevant properties are highly desirable to be preserved at the discrete level with the least time-step restrictions.
We first construct our schemes for a reduced PNP model, then extend to multi-dimensional PNP equations and a class of FP equations with interaction potentials. The common strategies in the construction of the baseline schemes include two ingredients: (i) reformulation of each underlying model so that the resulting system is more suitable for constructing positive schemes, and (ii) integration of semi-implicit time discretization and central spatial discretization. For each model equation, we show that the semi-discrete schemes (continuous in time) preserve all three solution properties (positivity, mass conservation, and energy dissipation). The fully discrete first order schemes preserve solution positivity and mass conservation for arbitrary time steps. Moreover, there exists a discrete energy function which dissipates along time marching with an bound on time steps.
We show that the second order (in both time and space) schemes preserve solution positivity for suitably small time steps; for larger time steps, we apply a local limiter to restore the solution positivity. We prove that such limiter preserves local mass and does not destroy the approximation accuracy. In addition, the limiter provides a reliable way of restoring solution positivity for other high order conservative finite difference or finite volume schemes.
Both the first and second order schemes are linear and can be efficiently implemented without resorting to any iteration method. The second order schemes are only slight modifications of the first order schemes. Computational costs of a single time step for first and second order schemes are similar, hence our second-order in time schemes are efficient than the first-order in time schemes, given a larger time step could be utilized (to save computational cost). We conduct extensive numerical tests that support our theoretical results and illustrate the accuracy, efficiency, and capacity to preserve the solution properties of our schemes
Efficient discontinuous Galerkin (DG) methods for time-dependent fourth order problems
In this thesis, we design, analyze and implement efficient discontinuous Galerkin (DG) methods for a class of fourth order time-dependent partial differential equations (PDEs). The main advantages of such schemes are their provable unconditional stability, high order accuracy, and their easiness for generalization to multi-dimensions for arbitrarily high order schemes on structured and unstructured meshes. These schemes have been applied to two fourth order gradient flows such as the Swift-Hohenberg (SH) equation and the Cahn-Hilliard (CH) equation, which are well known nonlinear models in modern physics.
For fourth order PDEs of the form , where is an adjoint elliptic operator, the fully discrete DG schemes are constructed in several steps: (a) rewriting the equation as a system of second order PDEs so that ; (b) applying the DG discretization to this mixed formulation with central numerical fluxes on interior interfaces and weakly enforcing the specified boundary conditions; and (c) combining a special class of time discretizations, that allows
the method to be unconditionally stable regardless of its accuracy.
Main contributions of this thesis are as follows:
Firstly, we introduce mixed discontinuous Galerkin methods without interior penalty for the spatial DG discretization, and the semi-discrete schemes are shown
stable for linear problems, and unconditionally energy stable for nonlinear gradient flows. For the mixed DG method applied to linear problems with
periodic boundary conditions, we establish the optimal error estimate of order for polynomials of degree with the
Crank-Nicolson time discretization. In addition, the resulting DG methods can easily handle different boundary conditions.
Secondly, for a class of fourth order gradient flow problems, including the SH equation, we combine the so-called \emph{Invariant Energy Quadratization} (IEQ) approach [X. Yang, J. Comput. Phys., 327:294{316, 2016] as time discretization. Coupled with a projection step for the auxiliary variable, both first and second order EQ-DG schemes are shown unconditionally energy stable. In addition, they are linear and can be efficiently solved without resorting to any iteration method. We present extensive numerical examples that support our theoretical results and illustrate the efficiency, accuracy, and stability of our new algorithms. Benchmark problems are also presented to examine the long time behavior of the numerical solutions.
Both the theoretical and algorithmic aspects of these methods have potentially wide applications. Progress is made with the IEQ-DG framework to solve the Cahn-Hilliard equation. With the usual penalty in the DG discretization, the resulting EQ-DG schemes are shown to be able to produce free-energy-decaying, and mass conservative solutions, irrespective of the time step and the mesh size. In addition, the schemes are easy to implement, and test cases for the Cahn-Hilliard equation will be reported
Fast iterative solvers for Cahn-Hilliard problems
Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Dissertation, 2016von M. Sc. Jessica BoschLiteraturverzeichnis: Seite [247]-25