Abstract: We present an unconditionally energy stable finite-difference scheme for the phase field crystal equation. The method is based on a convex splitting of a discrete energy and is semi-implicit. The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable, regardless of time step size. We present local-in-time error estimates that ensure the convergence of the scheme. While this paper is primarily concerned with the phase field crystal equation, most of the theoretical results hold for the related Swift-Hohenberg equation as well.

Abstract: This paper describes the formulation, verification, and validation of a depth-integrated, non-hydrostatic model with a semi-implicit, finite difference scheme. The formulation builds on the nonlinear shallow-water equations and utilizes a non-hydrostatic pressure term to describe weakly dispersive waves. A momentum-conserved advection scheme enables modeling of breaking waves without the aid of analytical solutions for bore approximation or empirical equations for energy dissipation. An upwind scheme extrapolates the free-surface elevation instead of the flow depth to provide the flux in the momentum and continuity equations. This greatly improves the model stability, which is essential for computation of energetic breaking waves and run-up. The computed results show very good agreement with laboratory data for wave propagation, transformation, breaking, and run-up. Since the numerical scheme to the momentum and continuity equations remains explicit, the implicit non-hydrostatic solution is directly applicable to existing nonlinear shallow-water models. Copyright © 2008 John Wiley & Sons, Ltd.

Abstract: We developed a mimetic finite difference method for solving elliptic equations with tensor coefficients on polyhedral meshes. The first-order convergence estimates in a mesh-dependent H 1 norm are derived.

Abstract: A dimension reduction method called Discrete Empirical Interpolation is proposed and shown to dramatically reduce the computational complexity of the popular Proper Orthogonal Decomposition (POD) method for constructing reduced-order models for unsteady and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. Empirical Interpolation posed in finite dimensional function space is a modification of POD that reduces complexity of the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a Discrete Empirical Interpolation Method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of unsteady time dependent PDE and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of EIM in a finite dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the 1-D FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captured non-linear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

Abstract: A method for solving boundary value problems (BVPs) is introduced using artificial neural networks (ANNs) for irregular domain boundaries with mixed Dirichlet/Neumann boundary conditions (BCs). The approximate ANN solution automatically satisfies BCs at all stages of training, including before training commences. This method is simpler than other ANN methods for solving BVPs due to its unconstrained nature and because automatic satisfaction of Dirichlet BCs provides a good starting approximate solution for significant portions of the domain. Automatic satisfaction of BCs is accomplished by the introduction of an innovative length factor. Several examples of BVP solution are presented for both linear and nonlinear differential equations in two and three dimensions. Error norms in the approximate solution on the order of 10-4 to 10-5 are reported for all example problems.

Abstract: Preface. Introduction. 1. Theory of differential equations: an introduction. 1.1 General solvability theory. 1.2 Stability of the initial value problem. 1.3 Direction fields. Problems. 2. Euler's method. 2.1 Euler's method. 2.2 Error analysis of Euler's method. 2.3 Asymptotic error analysis. 2.3.1 Richardson extrapolation. 2.4 Numerical stability. 2.4.1 Rounding error accumulation. Problems. 3. Systems of differential equations. 3.1 Higher order differential equations. 3.2 Numerical methods for systems. Problems. 4. The backward Euler method and the trapezoidal method. 4.1 The backward Euler method. 4.2 The trapezoidal method. Problems. 5. Taylor and Runge-Kutta methods. 5.1 Taylor methods. 5.2 Runge-Kutta methods. 5.3 Convergence, stability, and asymptotic error. 5.4 Runge-Kutta-Fehlberg methods. 5.5 Matlab codes. 5.6 Implicit Runge-Kutta methods. Problems. 6. Multistep methods. 6.1 Adams-Bashforth methods. 6.2 Adams-Moulton methods. 6.3 Computer codes. Problems. 7. General error analysis for multistep methods. 7.1 Truncation error. 7.2 Convergence. 7.3 A general error analysis. Problems. 8. Stiff differential equations. 8.1 The method of lines for a parabolic equation. 8.2 Backward differentiation formulas. 8.3 Stability regions for multistep methods. 8.4 Additional sources of difficulty. 8.5 Solving the finite difference method. 8.6 Computer codes. Problems. 9. Implicit RK methods for stiff differential equations. 9.1 Families of implicit Runge-Kutta methods. 9.2 Stability of Runge-Kutta methods. 9.3 Order reduction. 9.4 Runge-Kutta methods for stiff equations in practice. Problems. 10. Differential algebraic equations. 10.1 Initial conditions and drift. 10.2 DAEs as stiff differential equations. 10.3 Numerical issues: higher index problems. 10.4 Backward differentiation methods for DAEs. 10.5 Runge-Kutta methods for DAEs. 10.6 Index three problems from mechanics. 10.7 Higher index DAEs. Problems. 11. Two-point boundary value problems. 11.1 A finite difference method. 11.2 Nonlinear two-point boundary value problems. Problems. 12. Volterra integral equations. 12.1 Solvability theory. 12.2 Numerical methods. 12.3 Numerical methods - Theory. Problems. Appendix A. Taylor's theorem. Appendix B. Polynomial interpolation. Bibliography. Index.

Abstract: SUMMARY We derive explicit and new implicit staggered-grid finite-difference (FD) formulas for derivatives of first order with any order of accuracy by a plane wave theory and Taylor’s series expansion. Furthermore, we arrive at a practical algorithm such that the tridiagonal matrix equations are formed by the implicit FD formulas derived from the fractional expansion of derivatives. Our results demonstrate that the accuracy of a (2N + 2)th-order implicit formula is nearly equivalent to or greater than that of a (4N)th-order explicit formula. The new implicit method only involves solving tridiagonal matrix equations. We also demonstrate that a( 2N + 2)th-order implicit formulation requires nearly the same amount of memory and computation as those of a (2N + 4)th-order explicit formulation but attains the accuracy achieved by a (4N)th-order explicit formulation when additional cost of visiting arrays is not considered. Our analysis of efficiency and numerical modelling results for elastic wave propagation demonstrates that a high-order explicit staggered-grid method can be replaced by an implicit staggered-grid method of some order, which will increase the accuracy but not the computational cost.

Abstract: Service life of concrete structures under chloride environment can be predicted by formulations based on the mechanism of chloride ion diffusion. This mechanism can be mathematically described using the partial differential equation (PDE) of the Fick’s second law. One-dimensional PDE can be solved analytically by assuming constant surface chloride ion concentration and constant diffusion coefficient. However, the solution becomes more complicated when two additional conditions are included, i.e., concrete cover repair or replacement and time dependent variation of the surface chloride ion concentration and diffusion coefficient. In this paper, a numerical finite difference based formulation is proposed to effectively accommodate these two additional conditions. By virtue of numerical computation, the nonlinear initial chloride ion concentration can be treated in point-wise manner and both the time dependent surface chloride ion concentration and diffusion coefficient can be iteratively updated. Based on a Crank–Nicolson scheme within the finite difference method, a proper formulation accounting for space-dependent diffusion coefficient was derived; chloride ion concentration profiles are obtained and the service life of repaired concrete structures under chloride environment is predicted. Numerical examples and observations are finally presented.

Abstract: The key objective of this work is the design of an unconditionally stable, robust, efficient, modular, and easily expandable finite element-based simulation tool for cardiac electrophysiology. In contrast to existing formulations, we propose a global–local split of the system of equations in which the global variable is the fast action potential that is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level. Cell-specific excitation characteristics are thus strictly local and only affect the constitutive level. We illustrate the modular character of the model in terms of the FitzHugh–Nagumo model for oscillatory pacemaker cells and the Aliev–Panfilov model for non-oscillatory ventricular muscle cells. We apply an implicit Euler backward finite difference scheme for the temporal discretization and a finite element scheme for the spatial discretization. The resulting non-linear system of equations is solved with an incremental iterative Newton–Raphson solution procedure. Since this framework only introduces one single scalar-valued variable on the node level, it is extremely efficient, remarkably stable, and highly robust. The features of the general framework will be demonstrated by selected benchmark problems for cardiac physiology and a two-dimensional patient-specific cardiac excitation problem. Copyright © 2009 John Wiley & Sons, Ltd.

Abstract: Preface. List of Boxed Examples. 1 Panta Rei: Everything Flows. 1.1 Historical Perspective. 1.2 What Lies Ahead? 1.3 How Nature Uses RD. 1.3.1 Animate Systems. 1.3.2 Inanimate Systems. 1.4 RD in Science and Technology. References. 2 Basic Ingredients: Diffusion. 2.1 Diffusion Equation. 2.2 Solving Diffusion Equations. 2.2.1 Separation of Variables. 2.2.2 Laplace Transforms. 2.3 The Use of Symmetry and Superposition. 2.4 Cylindrical and Spherical Coordinates. 2.5 Advanced Topics. References. 3 Chemical Reactions. 3.1 Reactions and Rates. 3.2 Chemical Equilibrium. 3.3 Ionic Reactions and Solubility Products. 3.4 Autocatalysis, Cooperativity and Feedback. 3.5 Oscillating Reactions. 3.6 Reactions in Gels. References. 4 Putting It All Together: Reaction-Diffusion Equations and the Methods of Solving Them. 4.1 General Form of Reaction-Diffusion Equations. 4.2 RD Equations that can be Solved Analytically. 4.3 Spatial Discretization. 4.3.1 Finite Difference Methods. 4.3.2 Finite Element Methods. 4.4 Temporal Discretization and Integration. 4.4.1 Case 1: tau Rxn => tau Diff . 4.4.1.1 Forward Time Centered Space (FTCS) Differencing. 4.4.1.2 Backward Time Centered Space (BTCS) Differencing. 4.4.1.3 Crank-Nicholson Method. 4.4.1.4 Alternating Direction Implicit Method in Two and Three Dimensions. 4.4.2 Case 2: tau Rxn < tau Diff . 4.4.2.1 Operator Splitting Method. 4.4.2.2 Method of Lines. 4.4.3 Dealing with Precipitation Reactions. 4.5 Heuristic Rules for Selecting a Numerical Method. 4.6 Mesoscopic Models. References. 5 Spatial Control of Reaction-Diffusion at Small Scales: Wet Stamping (WETS). 5.1 Choice of Gels. 5.2 Fabrication. Appendix 5A: Practical Guide to Making Agarose Stamps. 5A.1 PDMS Molding. 5A.2 Agarose Molding. References. 6 Fabrication by Reaction-Diffusion: Curvilinear Microstructures for Optics and Fluidics. 6.1 Microfabrication: The Simple and the Difficult. 6.2 Fabricating Arrays of Microlenses by RD and WETS. 6.3 Intermezzo: Some Thoughts on Rational Design. 6.4 Guiding Microlens Fabrication by Lattice Gas Modeling. 6.5 Disjoint Features and Microfabrication of Multilevel Structures. 6.6 Microfabrication of Microfluidic Devices. 6.7 Short Summary. References. 7 Multitasking: Micro- and Nanofabrication with Periodic Precipitation. 7.1 Periodic Precipitation. 7.2 Phenomenology of Periodic Precipitation. 7.3 Governing Equations. 7.4 Microscopic PP Patterns in Two Dimensions. 7.4.1 Feature Dimensions and Spacing. 7.4.2 Gel Thickness. 7.4.3 Degree of Gel Crosslinking. 7.4.4 Concentration of the Outer and Inner Electrolytes. 7.5 Two-Dimensional Patterns for Diffractive Optics. 7.6 Buckling into the Third Dimension: Periodic 'Nanowrinkles'. 7.7 Toward the Applications of Buckled Surfaces. 7.8 Parallel Reactions and the Nanoscale. References. 8 Reaction-Diffusion at Interfaces: Structuring Solid Materials. 8.1 Deposition of Metal Foils at Gel Interfaces. 8.1.1 RD in the Plating Solution: Film Topography. 8.1.2 RD in the Gel Substrates: Film Roughness. 8.2 Cutting into Hard Solids with Soft Gels. 8.2.1 Etching Equations. 8.2.1.1 Gold Etching. 8.2.1.2 Glass and Silicon Etching. 8.2.2 Structuring Metal Films. 8.2.3 Microetching Transparent Conductive Oxides, Semiconductors and Crystals. 8.2.4 Imprinting Functional Architectures into Glass. 8.3 The Take-Home Message. References. 9 Micro-chameleons: Reaction-Diffusion for Amplification and Sensing. 9.1 Amplification of Material Properties by RD Micronetworks. 9.2 Amplifying Macromolecular Changes using Low-Symmetry Networks. 9.3 Detecting Molecular Monolayers. 9.4 Sensing Chemical 'Food'. 9.4.1 Oscillatory Kinetics. 9.4.2 Diffusive Coupling. 9.4.3 Wave Emission and Mode Switching. 9.5 Extensions: New Chemistries, Applications and Measurements. References. 10 Reaction-Diffusion in Three Dimensions and at the Nanoscale. 10.1 Fabrication Inside Porous Particles. 10.1.1 Making Spheres Inside of Cubes. 10.1.2 Modeling of 3D RD. 10.1.3 Fabrication Inside of Complex-Shape Particles. 10.1.4 'Remote' Exchange of the Cores. 10.1.5 Self-Assembly of Open-Lattice Crystals. 10.2 Diffusion in Solids: The Kirkendall Effect and Fabrication of Core-Shell Nanoparticles. 10.3 Galvanic Replacement and De-Alloying Reactions at the Nanoscale: Synthesis of Nanocages. References. 11 Epilogue: Challenges and Opportunities for the Future. References. Appendix A: Nature's Art. Appendix B: Matlab Code for the Minotaur (Example 4.1). Appendix C: C++ Code for the Zebra (Example 4.3). Index.

Abstract: I have developed and solved the constant- Q model for the attenuation of P- and S-waves in the time domain using a new modeling algorithm based on fractional derivatives. The model requires time derivatives of order m+2γ applied to the strain components, where m=0,1,… and γ= (1∕π) tan−1 (1∕Q) , with Q the P-wave or S-wave quality factor. The derivatives are computed with the Grunwald-Letnikov and central-difference fractional approximations, which are extensions of the standard finite-difference operators for derivatives of integer order. The modeling uses the Fourier method to compute the spatial derivatives, and therefore can handle complex geometries and general material-property variability. I verified the results by comparison with the 2D analytical solution obtained for wave propagation in homogeneous Pierre Shale. Moreover, the modeling algorithm was used to compute synthetic seismograms in heterogeneous media corresponding to a crosswell seismic experiment.

Abstract: Numerical modeling of seismic waves in heterogeneous, porous reservoir rocks is an important tool for interpreting seismic surveys in reservoir engineering. Various theoretical studies derive effective elastic moduli and seismic attributes from complex rock properties, involving patchy saturation and fractured media. To confirm and further develop rock-physics theories for reservoir rocks, accurate numerical modeling tools are required. Our 2D velocity-stress, finite-difference scheme simulates waves within poroelastic media as described by Biot’s theory. The scheme is second order in time, contains high-order spatial derivative operators, and is parallelized using the domain-decomposition technique. A series of numerical experiments that are compared to exact analytical solutions allow us to assess the stability conditions and dispersion relations of the explicit poroelastic finite-differ-ence method. The focus of the experiments is to model wave-induced flow accurately in the vicinity of mesoscopic hete...

Abstract: We simulate numerically the full dynamics of Faraday waves in three dimensions for two incompressible and immiscible viscous fluids. The Navier–Stokes equations are solved using a finite-difference projection method coupled with a front-tracking method for the interface between the two fluids. The critical accelerations and wavenumbers, as well as the temporal behaviour at onset are compared with the results of the linear Floquet analysis of Kumar & Tuckerman (J. Fluid Mech., vol. 279, 1994, p. 49). The finite-amplitude results are compared with the experiments of Kityk et al (Phys. Rev. E, vol. 72, 2005, p. 036209). In particular, we reproduce the detailed spatio-temporal spectrum of both square and hexagonal patterns within experimental uncertainty. We present the first calculations of a three-dimensional velocity field arising from the Faraday instability for a hexagonal pattern as it varies over its oscillation period.

Abstract: In this paper we present a three-dimensional Navier–Stokes solver for incompressible two-phase flow problems with surface tension and apply the proposed scheme to the simulation of bubble and droplet deformation. One of the main concerns of this study is the impact of surface tension and its discretization on the overall convergence behavior and conservation properties. Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin's projection approach. The free surface between the two fluid phases is tracked with a level set (LS) technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and a third-order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the LS function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second-order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface, which leads to a first-order convergence in the smoothing region. We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature, and the application of our solver to the simulation of two rising bubble problems, one with small and one with large jumps in material parameters, and the simulation of a droplet deformation due to a shear flow in three space dimensions. Furthermore, we compare our three-dimensional results with those of quasi-two-dimensional and two-dimensional simulations. This comparison clearly shows the need for full three-dimensional simulations of droplet and bubble deformation to capture the correct physical behavior. Copyright © 2009 John Wiley & Sons, Ltd.