The ADI-FDTD Method for High Accuracy Electrophysics Applications

The Finite-Difference Time-Domain (FDTD) is a dependable method to simulate a wide range of problems from acoustics, to electromagnetics, and to photonics, amongst others. The execution time of an FDTD simulation is inversely proportional to the time-step size. Since the FDTD method is explicit, its time-step size is limited by the well-known Courant-Friedrich-Levy (CFL) stability limit. The CFL stability limit can render the simulation inefficient for very fine structures. The Alternating Direction Implicit FDTD (ADI-FDTD) method has been introduced as an unconditionally stable implicit method. Numerous works have shown that the ADI-FDTD method is stable even when the CFL stability limit is exceeded. Therefore, the ADI-FDTD method can be considered an efficient method for special classes of problems with very fine structures or high gradient fields. Whenever the ADI-FDTD method is used to simulate open-region radiation or scattering problems, the implementation of a mesh-truncation scheme or absorbing boundary condition becomes an integral part of the simulation. These truncation techniques represent, in essence, differential operators that are discretized using a distinct differencing scheme which can potentially affect the stability of the scheme used for the interior region. In this work, we show that the ADI-FDTD method can be rendered unstable when higher-order mesh truncation techniques such as Higdon's Absorbing Boundary Condition (ABC) or Complementary Derivatives Method (COM) are used. When having large field gradients within a limited volume, a non-uniform grid can reduce the computational domain and, therefore, it decreases the computational cost of the FDTD method. However, for high-accuracy problems, different grid sizes increase the truncation error at the boundary of domains having different grid sizes. To address this problem, we introduce the Complementary Derivatives Method (CDM), a second-order accurate interpolation scheme. The CDM theory is discussed and applied to numerical examples employing the FDTD and ADI-FDTD methods

Volume 2: Explicit, multistage upwind schemes for Euler and Navier-Stokes equations

The objective of this study was to develop a high-resolution-explicit-multi-block numerical algorithm, suitable for efficient computation of the three-dimensional, time-dependent Euler and Navier-Stokes equations. The resulting algorithm has employed a finite volume approach, using monotonic upstream schemes for conservation laws (MUSCL)-type differencing to obtain state variables at cell interface. Variable interpolations were written in the k-scheme formulation. Inviscid fluxes were calculated via Roe's flux-difference splitting, and van Leer's flux-vector splitting techniques, which are considered state of the art. The viscous terms were discretized using a second-order, central-difference operator. Two classes of explicit time integration has been investigated for solving the compressible inviscid/viscous flow problems--two-state predictor-corrector schemes, and multistage time-stepping schemes. The coefficients of the multistage time-stepping schemes have been modified successfully to achieve better performance with upwind differencing. A technique was developed to optimize the coefficients for good high-frequency damping at relatively high CFL numbers. Local time-stepping, implicit residual smoothing, and multigrid procedure were added to the explicit time stepping scheme to accelerate convergence to steady-state. The developed algorithm was implemented successfully in a multi-block code, which provides complete topological and geometric flexibility. The only requirement is C degree continuity of the grid across the block interface. The algorithm has been validated on a diverse set of three-dimensional test cases of increasing complexity. The cases studied were: (1) supersonic corner flow; (2) supersonic plume flow; (3) laminar and turbulent flow over a flat plate; (4) transonic flow over an ONERA M6 wing; and (5) unsteady flow of a compressible jet impinging on a ground plane (with and without cross flow). The emphasis of the test cases was validation of code, and assessment of performance, as well as demonstration of flexibility

Quantum Variational Solving of Nonlinear and Multi-Dimensional Partial Differential Equations

A variational quantum algorithm for numerically solving partial differential equations (PDEs) on a quantum computer was proposed by Lubasch et al. In this paper, we generalize the method introduced by Lubasch et al. to cover a broader class of nonlinear PDEs as well as multidimensional PDEs, and study the performance of the variational quantum algorithm on several example equations. Specifically, we show via numerical simulations that the algorithm can solve instances of the Single-Asset Black-Scholes equation with a nontrivial nonlinear volatility model, the Double-Asset Black-Scholes equation, the Buckmaster equation, and the deterministic Kardar-Parisi-Zhang equation. Our simulations used up to $n=12$ ansatz qubits, computing PDE solutions with $2^n$ grid points. We also performed proof-of-concept experiments with a trapped-ion quantum processor from IonQ, showing accurate computation of two representative expectation values needed for the calculation of a single timestep of the nonlinear Black--Scholes equation. Through our classical simulations and experiments on quantum hardware, we have identified -- and we discuss -- several open challenges for using quantum variational methods to solve PDEs in a regime with a large number ($\gg 2^{20}$) of grid points, but also a practical number of gates per circuit and circuit shots.Comment: 16 pages, 10 figures (main text

A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also

Magnetic Drug Targeting: Developing the Basics

Focusing medicine to disease locations is a needed ability to treat a variety of pathologies. During chemotherapy, for example, typically less than 0.1% of the drugs are taken up by tumor cells, with the remaining 99.9% going into healthy tissue. Physicians often select the dosage by how much a patient can physically withstand rather than by how much is needed to kill all the tumor cells. The ability to actively position medicine, to physically direct and focus it to specific locations in the body, would allow better treatment of not only cancer but many other diseases. Magnetic drug targeting (MDT) harnesses therapeutics attached to magnetizable particles, directing them to disease locations using magnetic fields. Particles injected into the vasculature will circulate throughout the body as the applied magnetic field is used to attempt confinement at target locations. The goal is to use the reservoir of particles in the general circulation and target a specific location by pulling the nanoparticles using magnetic forces. This dissertation adds three main advancements to development of magnetic drug targeting. Chapter 2 develops a comprehensive ferrofluid transport model within any blood vessel and surrounding tissue under an applied magnetic field. Chapter 3 creates a ferrofluid mobility model to predict ferrofluid and drug concentrations within physiologically relevant tissue architectures established from human autopsy samples. Chapter 4 optimizes the applied magnetic fields within the particle mobility models to predict the best treatment scenarios for two classes of chemotherapies for treating future patients with hepatic metastatic breast cancer microtumors

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

Matrix-free finite-element computations at extreme scale and for challenging applications

For numerical computations based on finite element methods (FEM), it is common practice to assemble the system matrix related to the discretized system and to pass this matrix to an iterative solver. However, the assembly step can be costly and the matrix might become locally dense, e.g., in the context of high-order, high-dimensional, or strongly coupled multicomponent FEM, leading to high costs when applying the matrix due to limited bandwidth on modern CPU- and GPU-based hardware. Matrix-free algorithms are a means of accelerating FEM computations on HPC systems, by applying the effect of the system matrix without assembling it. Despite convincing arguments for matrix-free computations as a means of improving performance, their usage still tends to be an exception at the time of writing of this thesis, not least because they have not yet proven their applicability in all areas of computational science, e.g., solid mechanics. In this thesis, we further develop a state-of-the-art matrix-free framework for high-order FEM computations with focus on the preconditioning and adopt it in novel application fields. In the context of high-order FEM, we develop means of improving cache efficiency by interleaving cell loops with vector updates, which we use to increase the throughput of preconditioned conjugate gradient methods and of block smoothers based on additive Schwarz methods; we also propose an algorithm for the fast application of hanging-node constraints in 3D for up to 137 refinement configurations. We develop efficient geometric and polynomial multigrid solvers with optimized transfer operators, whose performance is experimentally investigated in detail in the context of locally refined meshes, indicating the superiority of global-coarsening algorithms. We apply the developed solvers in the context of novel stage-parallel implicit Runge–Kutta methods and demonstrate the benefit of stage–parallel solvers in decreasing the time to solution at the scaling limit. Novel challenging application fields of matrix-free computations include high-dimensional computational plasma physics, solid-state-sintering simulations with a high and dynamically changing number of strongly coupled components, and coupled multiphysics problems with evaluation and integration at arbitrary points. In the context of these fields, we detail computational challenges, propose modified versions of the standard matrix-free algorithms for high-performance computing, and discuss preconditioning-related topics. The efficiency of the derived algorithms on the node level and at extreme scales is demonstrated experimentally on SuperMUC-NG, one of Germany’s leading supercomputers, with up to 150k processes and by solving systems of up to 5 × 1012 unknowns. Such problem sizes would not be conceivable for equivalent matrix-based algorithms. The major achievements of this thesis allow to run larger simulations faster and more efficiently, enabling progress and new possibilities for a range of application fields in computational science

Exponential integrators: tensor structured problems and applications

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed