Explicit Marching Scheme Research Articles

Explicit unconditionally stable methods for the heat equation via potential theory

We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite difference or finite element schemes for the heat equation are stable only if the time step $\Delta t$ is of the order $O(\Delta x^2)$, where $\Delta x$ is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions $d\ge 1$, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an $L^2$-norm of the solution to the integral equation is bounded by $c_dT^{d/2}$ times the norm of the right hand side. For the Robin problem on the half space in any dimension, with constant Robin (heat transfer) coefficient $\kappa$, we exhibit a constant $C$ such that the forward Euler scheme is stable if $\Delta t < C/\kappa^2$, independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in $L^\infty$-norm.

Stabilized backward in time explicit marching schemes in the numerical computation of ill-posed time-reversed hyperbolic/parabolic systems

ABSTRACTThis paper develops stabilized explicit marching difference schemes that can successfully solve a significant but limited class of multidimensional, ill-posed, backward in time problems for coupled hyperbolic/parabolic systems associated with vibrating thermoelastic plates and coupled sound and heat flow. Stabilization is achieved by applying compensating smoothing operators at each time step, to quench the instability. Analysis of convergence is restricted to the transparent case of linear, autonomous, selfadjoint spatial differential operators, and almost best-possible error bounds are obtained for backward in time reconstruction in that class of problems. However, the actual computational schemes can be applied to more general problems, including examples with variable time dependent coefficients, as well as nonlinearities. The stabilized explicit schemes are unconditionally stable, marching forward or backward in time, but the smoothing operation at each step leads to a distortion away from the true solution. This is the stabilization penalty. It is shown that in many problems of interest, that distortion is small enough to allow for useful results. Backward in time continuation is illustrated using pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Several computational experiments show that efficient FFT-synthesized smoothing operators, based on with real , can be successfully applied in a broad range of problems.

Stable explicit stepwise marching scheme in ill-posed time-reversed viscous wave equations

The numerical computation of ill-posed, nonlinear, multidimensional initial value problems presents considerable difficulties. Conventional stepwise marching schemes for such problems, whether explicit or implicit, are necessarily unconditionally unstable and result in explosive noise amplification. Following previous work on backward parabolic equations, this paper develops and analyses a stabilized explicit marching scheme for ill-posed time-reversed viscous wave equations. The method uses easily synthesized linear smoothing operators at each time step to quench the instability. Smoothing operators based on positive real powers p of the negative Laplacian are helpful, and can be realized efficiently on rectangular domains using FFT algorithms. The stabilized explicit scheme is unconditionally stable, marching forward or backward in time, and can be applied to nonlinear viscous wave equations by simply lagging the nonlinearity at the previous time step. However, the smoothing operation at each step leads to a distortion away from the true solution. This is the stabilization penalty. It is shown that in many problems of interest, that distortion is often small enough to allow for useful results. In the canonical case of linear autonomous selfadjoint time-reversed viscous wave equations, with solutions satisfying prescribed bounds, it is proved that the stabilized explicit scheme leads to an error estimate differing from the best-possible estimate, only by the stabilization penalty. The procedure is a valuable complement to the well-known quasi-reversibility method. As illustrative examples, the paper uses fictitiously blurred pixel images, obtained using sharp images as initial values in linear or nonlinear well-posed, forward viscous wave equations. Such images are associated with highly irregular underlying data intensity surfaces that can severely challenge reconstruction procedures. Deblurring these images proceeds by applying the stabilized explicit scheme on the corresponding ill-posed, time-reversed equation. Instructive computational experiments demonstrate the capabilities of the method on 2D rectangular regions.

Time reversal with partial information for wave refocusing and scatterer identification

Time reversal is a well-known procedure in application fields involving wave propagation. Among other uses, it can be applied as a computational tool for solving certain inverse problems. The procedure is based on advancing the solution of the relevant wave problem “backward in time”. One important use of numerical time-reversal is that of refocusing, where a reverse run is performed to recover the location of a source applied at an initial time based on measurements at a later time. Usually, only partial, noisy, information is available, at certain measurement locations, on the field values that serve as data for the reverse run. In this paper, the question concerning the amount and characterization of the available data needed for a successful refocusing is studied for the scalar wave equation. In particular, a simple procedure is proposed which exploits multiple measurement times, and is shown to be very beneficial for refocusing. A tradeoff between availability of spatial and temporal information is discussed. The effect of measurement noise is studied, and the technique is shown to be quite robust, sometimes even in the presence of very high noise levels. The use of the technique as a basis for scatterer identification is also discussed. A numerical study of these effects is presented, employing finite elements in space and a standard explicit marching scheme in time. In contrast to some previous studies, the propagation medium is taken to be homogeneous.

High order marching schemes for the wave equation in complex geometry

We present a new class of explicit marching schemes for the wave equation in complex geometry. They rely on a simple embedding of the domain in a uniform Cartesian grid, which allows for efficient and automatic implementation but creates irregular cells near the boundary. While existing explicit finite difference schemes are generally restricted in the size of the time step that can be taken by the dimensions of the smallest cell, the schemes described here are capable of taking time steps dictated by the uniform grid spacing. This should be of significant benefit in a wide variety of simulation efforts.

Strongly consistent marching schemes for the wave equation

In this paper, we consider a class of explicit marching schemes first proposed in [1] for solving the wave equation in complex geometry using an embedded Cartesian grid. These schemes rely on an integral evolution formula for which the numerical domain of dependence adjusts automatically to contain the true domain of dependence. Here, we refine and analyze a subclass of such schemes, which satisfy a condition we refer to as strong u-consistency. This requires that the evolution scheme be exact for a single-valued approximation to the solution at the previous time steps. We provide evidence that many of these strongly u-consistent schemes are stable and converge at very high order even in the presence of small cells in the grid, while taking time steps dictated by the uniform grid spacing.

Space—time CESE method for the convection equation

In the paper, we study the analytic convergence of the space-time conservation element and solution element method for smooth solutions of the convection equation. Stability of the method is limited by the CFL condition and a parameter,e,which controls the numerical dissipation. We show that the method converges under appropriate conditions on the mesh and on the parameter,e, by considering a one-dimensional convection equation. The main advantage of this second order scheme is that it is an explicit marching scheme that allows one to solve for both the function and its derivative at the same time with comparable accuracy.Numerical simulations are presented to verify the convergence results for both the linear convection equation and a non-linear conservation law.

A Dynamic Mesh Algorithm for Curvature Dependent Evolving Interfaces

A new finite element method is discussed for approximating evolving interfaces in2nwhose normal velocity equals mean curvature plus a forcing function. The method is insensitive to singularity formation and retains the local structure of the limit problem and, thus, exhibits a computational complexity typical of3n−1without having the drawbacks of front-tracking strategies. A graded dynamic mesh around the propagating front is the sole partition present at any time step and is significantly smaller than a full mesh. Time stepping is explicit, but stability constraints force small time steps only when singularities develop, whereas relatively large time steps are allowed before or past singularities, when the evolution is smooth. The explicit marching scheme also guarantees that at most one layer of elements has to be added or deleted per time step, thereby making mesh updating simple and, thus, practical. Performance and potentials are fully documented via a number of numerical simulations in 2D, 3D, 4D, and 8D, with axial symmetries. They include tori and cones for the mean curvature flow, minimal and prescribed mean curvature surfaces with given boundary, fattening for smooth driving force, and volume constraint.

Design of optimally smoothing multistage schemes for the euler equations

AbstractA recently derived local preconditioning of the Euler equations is shown to be useful in developing multistage schemes suited for multigrid use. The effect of the preconditioning matrix on the spatial Euler operator is to equalize the characteristic speeds. When applied to the discretized Euler equations, the preconditioning has the effect of strongly clustering the operator's eigenvalues in the complex plane. This makes possible the development of explicit marching schemes that effectively damp most high‐frequency Fourier modes, as desired in multigrid applications. The technique is the same as developed earlier for scalar convection schemes: placement of the zeros of the amplification factor of the multistage scheme in locations where eigenvalues corresponding to high‐frequency modes abound.

Modeling of Reactive Filling in Complex Cavities***

Abstract The flow of reacting monomers into thin planar cavities of irregular geometry is analyzed. The Hele-Shaw approximations are used to simplify the momentum and mass balances. At each time step, the flow domain is mapped onto a rectangular domain using elliptic expressions. The transformed governing equations are solved using finite-differences. The transient three-dimensional energy and species balances are solved using an explicit marching scheme coupled to the flow. Viscosity is modeled as a function of temperature and extent of reaction. Castro's model is used to compare several fountain flow approximations. A simplified algorithm was found to capture the essence of the fountain flow kinematics. The fountain flow approximations described here can also be used in thermoplastic injection molding modeling. Results for a rectangular end gated mold and for an irregular mold geometry are presented.