High-order numerical methods for 2D parabolic problems in single and composite domains

In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin

Residual, restarting and Richardson iteration for the matrix exponential

A well-known problem in computing some matrix functions iteratively is a lack of a clear, commonly accepted residual notion. An important matrix function for which this is the case is the matrix exponential. Assume, the matrix exponential of a given matrix times a given vector has to be computed. We interpret the sought after vector as a value of a vector function satisfying the linear system of ordinary differential equations (ODE), whose coefficients form the given matrix. The residual is then defined with respect to the initial-value problem for this ODE system. The residual introduced in this way can be seen as a backward error. We show how the residual can efficiently be computed within several iterative methods for the matrix exponential. This completely resolves the question of reliable stopping criteria for these methods. Furthermore, we show that the residual concept can be used to construct new residual-based iterative methods. In particular, a variant of the Richardson method for the new residual appears to provide an efficient way to restart Krylov subspace methods for evaluating the matrix exponential.\u

Global Discrete Artificial Boundary Conditions for Time-dependent Wave Propagation

We construct global artificial boundary conditions (ABCs) for the numerical simulation of wave processes on unbounded domains using a special non-deteriorating algorithm that has been developed previously for the long-term computation of wave-radiation solutions. The ABCs are obtained directly for the discrete formulation of the problem; in so doing, neither a rational approximation of "non-reflecting kernels," nor discretization of the continuous boundary conditions is required. The extent of temporal nonlocality of the new ABCs appears fixed and limited; in addition, the ABCs can handle artificial boundaries of irregular shape on regular grids with no fitting or adaptation needed and no accuracy loss induced

Difference Potentials and their Applications

In this lecture, we introduce the concept of difference potentials with the density from the space of discontinuities or jumps, which extends and generalizes the previous constructions of difference potentials; this new concept is sufficiently universal and at the same time simple. The apparatus of difference potentials constitutes the foundation of the difference potentials method (DPM). Before considering the actual constructions of difference potentials, we discuss some new opportunities that the DPM provides for computations. This brief introductory discussion (as well as the main part of the lecture itself) has a goal of drawing the attention of the scientific computing research community to the DPM and its applications. Although the construction of difference potentials with the density from the space of jumps presents an independent mathematical interest, the subject of this lecture will seem too abstract without discussing the possible applications in the beginning. Moreover, in the ..

Active Shielding and Control of Environmental Noise

We present a mathematical framework for the active control of time-harmonic acoustic disturbances. Unlike many existing methodologies, our approach provides for the exact volumetric cancellation of unwanted noise in a given predetermined region of space while leaving unaltered those components of the total acoustic field that are deemed as friendly. Our key finding is that for eliminating the unwanted component of the acoustic field in a given area, one needs to know relatively little; in particular, neither the locations nor structure nor strength of the exterior noise sources needs to be known. Likewise, there is no need to know the volumetric properties of the supporting medium across which the acoustic signals propagate, except, maybe, in the narrow area of space near the boundary (perimeter) of the domain to be shielded. The controls are built based solely on the measurements performed on the perimeter of the region to be shielded; moreover, the controls themselves (i.e., additional sources) are concentrated also only near this perimeter. Perhaps as important, the measured quantities can refer to the total acoustic field rather than to its unwanted component only, and the methodology can automatically distinguish between the two. In the paper, we construct a general solution to the aforementioned noise control problem. The apparatus used for deriving the general solution is closely connected to the concepts of generalized potentials and boundary projections of Calderon's type. For a given total wave field, the application of Calderon's projections allows us to definitively split between its incoming and outgoing components with respect to a particular domain of interest, which may have arbitrary shape. Then, the controls are designed so that they suppress the inc..