2D Continuous Galerkin Methods for Hyperbolic Equations

Parallel Sturmian comparison theorems are proved for hyperbolic and elliptic linear differential equations of second order, with special emphasis on cylindrical domains with the time axis parallel to the axis of the cylinder. These theorems have the following form: If a boundary problem I for a partial differential equation in a bounded domain has a nontrivial solution, then every solution of a second boundary problem II of the same type has a zero in the domain if the coefficient functions and boundary functions of II majorize those of I. The results extend and unify earlier results given for either hyperbolic equations or elliptic equations. In particular, the majorization hypotheses in Theorem 3, in the form of inequalities between the data functions in the differential equations and in the boundary conditions, indicate parallel sets of sufficient conditions for the Sturmian conclusion in the elliptic and hyperbolic cases. Counterexamples are given to show that two known theorems in the elliptic case do not extend to the hyperbolic case. Extensions to singular boundaries and mixed problems are included. The dichotomy referred to in the title pertains to the different but parallel Sturmian theories for elliptic and hyperbolic boundary problems: Results of similar type are true for both types of problems and yet other results true in the elliptic case are false in the hyperbolic case.

Mass conservation of the unified continuous and discontinuous element-based Galerkin methods on dynamically adaptive grids with application to atmospheric simulations

Analysis of a continuous Galerkin method with mesh modification for two-dimensional telegraph equation

This dissertation, framed within an Industrial Doctorate programme, has the twofold objective of improving the understanding of numerical properties and convergence acceleration techniques for High-Order methods, and to formulate an efficient incompressible flow solver for thermal applications. The first part of this manuscript is devoted to the study of the linear multi-grid method applied to Flux Reconstruction, a relatively recent High-Order numerical method with close relations to the well-known Discontinuous Galerkin and Spectral Difference schemes. We do so by investigating the non-modal dissipation behaviour of the multi-grid scheme as a function of its parameters, in an effort to find the most efficient configurations. The analysis is extended to 2D high aspect-ratio cells, in the presence of dominating convection. The second part of this manuscript introduces a High-Order incompressible flow solver devised by the simultaneous use of the Flux Reconstruction, the Discontinuous Galerkin and the Continuous Galerkin methods, each of which is known to offer increased computational efficiency for certain types of problems. On the one end of the spectrum, the Flux Reconstruction is an efficient algorithm for the explicit evaluation of fluxes in conservative form; on the other end, the Continuous Galerkin method is comparatively very efficient for the resolution of diffusion-dominated steady problems. The different formulations are put together in a fractional-step approach and validated against known test cases – unsteady Stokes and Taylor problems, lid-driven cavity, Taylor-Green vortex and thermally-driven cavity – with and without analytical solution, demonstrating convergence with mesh, time-step and polynomial refinement. RESUMEN El presente trabajo, enmarcado en un programa de Doctorado Industrial, tiene el doble objetivo de mejorar la comprensión de las propiedades numéricas y técnicas de aceleración de convergencia para métodos de alto orden, así como de formular un solver eficiente para la resolución de flujos incompresibles con aplicaciones térmicas. La primera parte del manuscrito está dedicada al estudio del método multi-grid lineal aplicado a Flux Reconstruction, un método numérico de alto orden relativamente reciente y estrechamente relacionado con los conocidos Discontinuous Galerkin y Spectral Difference. Ello se consigue investigando la disipación numérica no-modal del esquema multi-grid en función de sus parámetros, con el objetivo de determinar aquellas configuraciones más eficientes. El análisis se extiende a 2D con elementos con alta relación de aspecto en presencia de convección dominante. La segunda parte del manuscrito presenta un solver de alto orden incompresible concebido con el uso simultáneo de los métodos Flux Reconstruction, Discontinuous Galerkin y Continuous Galerkin, cada uno de los cuales ofrece una eficiencia computacional superior para ciertos tipos de problemas. En un extremo, Flux Reconstruction es un algoritmo eficiente para la evaluación explícita de flujos en forma conservativa; en el otro extremo, Continuous Galerkin es comparativamente muy eficiente para la resolución de problemas elípticos en estado estacionario. Las diferentes formulaciones se conjugan en un método de paso fraccionado validado con conocidos casos test con y sin solución analítica, demostrando convergencia con el refinamiento en el mallado, el orden polinomial y el paso de tiempo.

In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

Introduction Generalized Transport Phenomena Approach to Problem Analysis General Content Conservation Laws and Field Equations Concentrations, Velocities, and Fluxes Thermodynamics of Purely Viscous Fluid Mixtures Conservation of Mass for a One-Component System Conservation of Mass for a Mixture Modification of Field Equations for Mass Transfer Conservation of Linear Momentum for One-Component Systems Conservation of Linear Momentum for a Mixture Conservation of Moment of Momentum for One-Component Systems Conservation of Moment of Momentum for a Mixture Strategies for the Solution of Mass Transfer Problems Boundary Conditions Definitions Jump Balances for Mass Conservation Jump Balances for Linear Momentum Conservation Postulated Boundary Conditions at Phase Interfaces Boundary Conditions in the Absence of Mass Transfer Utilization of Jump Balances Additional Comments on Boundary Conditions Boundary Conditions and Uniqueness of Solutions Constitutive Equations Constitutive Principles First-Order Theory for Binary Systems Combined Field and Constitutive Equations for First-Order Binary Theory First-Order Theory for Ternary Systems Special Second-Order Theory for Binary Systems Viscoelastic Effects in Flow and Diffusion Validity of Constitutive Equations Parameters in Constitutive Equations General Approach in Parameter Determination Diffusion in Polymer-Solvent Mixtures Diffusion in Infinitely Dilute Polymer Solutions Diffusion in Dilute Polymer Solutions Diffusion in Concentrated Polymer Solutions - Free-Volume Theory for Self-Diffusion Diffusion in Concentrated Polymer Solutions - Mutual Diffusion Process Diffusion in Crosslinked Polymers Additional Properties of Diffusion Coefficients Special Behaviors of Polymer-Penetrant Systems Volumetric Behavior of Polymer-Penetrant Systems Sorption Behavior of Polymer-Penetrant Systems Antiplasticization Nonequilibrium at Polymer-Penetrant Interfaces Mathematical Apparatus Basic Definitions Classification of Second-Order Partial Differential Equations Specification of Boundary Conditions Sturm-Liouville Theory Series and Integral Representations of Functions Solution Methods for Partial Differential Equations Separation of Variables Method Separation of Variables Solutions Integral Transforms Similarity Transformations Green's Functions for Ordinary Differential Equations Green's Functions for Elliptic Equations Green's Functions for Parabolic Equations Perturbation Solutions Weighted Residual Method Solution Strategy for Mass Transfer Problems Proposed Solution Methods Induced Convection Solutions of a General Set of Mass Transfer Problems Mixing of Two Ideal Gases Steady Evaporation of a Liquid in a Tube Unsteady-State Evaporation Analysis of Free Diffusion Experiments Dissolution of a Rubbery Polymer Bubble Growth from Zero Initial Size Stability Behavior and Negative Concentrations in Ternary Systems Analysis of Impurity Migration in Plastic Containers Efficiency of Green's Function Solution Method Mass Transfer in Tube Flow Time-Dependent Interfacial Resistance Laminar Liquid Jet Diffusion Analysis Analysis of the Diaphragm Cell Dissolved Organic Carbon Removal from Marine Aquariums Unsteady Diffusion in a Block Copolymer Drying of Solvent-Coated Polymer Films Flow and Diffusion Past a Flat Plate with Solid Dissolution Gas Absorption in Vertical Laminar Liquid Jet Utilization of Polymers in Drug Delivery Gas Absorption and Diffusion into a Falling Liquid Film Perturbation Solutions of Mass Transfer Moving Boundary Problems Dissolution of a Plane Surface of a Pure Gas Phase Bubble Dissolution Singular Perturbations in Moving Boundary Problems Dropping Mercury Electrode Sorption in Thin Films Numerical Analysis of Mass Transfer Moving Boundary Problems Diffusion and Reaction Design of a Tubular Polymerization Reactor Transport Effects in Low-Pressure CVD Reactors Solution of Reaction Problems with First-Order Reactions Plug Flow Reactors with Variable Mass Density Bubble Dissolution and Chemical Reaction Danckwerts Boundary Conditions for Chemical Reactors Transport in Nonporous Membranes Assumptions Used in the Theory for Membrane Transport Steady Mass Transport in Binary Membranes Steady Mass Transport in Ternary Membranes Unsteady Mass Transport in Binary Membranes Phase Inversion Process for Forming Asymmetric Membranes Pressure Effects in Membranes Analysis of Sorption and Desorption Derivation of a Short-Time Solution Form for Sorption in Thin Films Sorption to a Film from a Pure Fluid of Finite Volume A General Analysis of Sorption in Thin Films Analysis of Step-Change Sorption Experiments Integral Sorption in Glassy Polymers Integral Sorption in Rubbery Polymers Oscillatory Diffusion and Diffusion Waves Dispersion and Chromatography Formulation of Taylor Dispersion Problem Dispersion in Laminar Tube Flow for Low Peclet Numbers Dispersion in Laminar Tube Flow for Long Times Dispersion in Laminar Tube Flow for Short Times Analysis of an Inverse Gas Chromatography Experiment Effects of Pressure Gradients on Diffusion: Wave Behavior and Sedimentation Wave Propagation in Binary Fluid Mixtures Hyperbolic Waves Dispersive Waves Time Effects for Parabolic and Hyperbolic Equations Sedimentation Equilibrium Viscoelastic Diffusion Experimental Results for Sorption Experiments Viscoelastic Effects in Step-Change Sorption Experiments Slow Bubble Dissolution in a Viscoelastic Fluid Transport with Moving Reference Frames Relationships Between Fixed and Moving Reference Frames Field Equations in Moving Reference Frames Steady Diffusion in an Ultracentrifuge Material Time Derivative Operators Frame Indifference of Material Time Derivatives Frame Indifference of Velocity Gradient Tensor Rheological Implications Appendix: Vector and Tensor Notation General Notation Conventions Vectors Tensors Results for Curvilinear Coordinates Material and Spatial Representations Reynolds' Transport Theorem

Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.

Practically speaking, fluid flow in round and noncircular nozzle is a very regular occurrence. The cold and hot water used in our homes is delivered to us via pipes. Water is delivered throughout the city via large pipe networks. Large pipelines carry natural gas and oil hundreds of kilometres from one place to another. During the operation of an engine, cooling water is carried by hoses to the radiator’s pipes, in which it is cooled as it travels. Experimentally, we detected the results of the model because there is no restriction on the application of experimental research to a certain sector or kind of concept. It may be used for a broad range of events and circumstances. Under parabolic velocity conditions, fluid (Water Pr 6.9) flows from the inlet position. The top and bottom walls of the rectangular nozzle are also moving at the same velocity as they are at the inlet position. Due to the movement of the walls, fluid is compressed in the particular region and also exhibits the same parabolic behavior. The solution of the coupled equations is determined by using the Finite Volume Method (FVM). When partial differential equations are expressed as algebraic equations, the FVM may be used to evaluate them. It can be used to evaluate elliptic, parabolic, and hyperbolic partial differential equations. Using FVM, it is necessary to know the values (and derivatives) of multiple variables at the cell faces, when the values (and derivatives) of these variables are only known at the cell centres. When determining these variables for convective terms, it is common to take the direction of the flow into consideration. The numerical results of the velocity and the pressure could be seen in the rectangular nozzle.

The work presents the analysis of the free boundary value problem related to the one-dimensional invasion model of new species in biofilm reactors. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of a system of non-linear hyperbolic partial differential equations governing the microbial species growth and a system of semi-linear elliptic partial differential equations describing the substrate trends. The model is completed with a system of elliptic partial differential equations governing the diffusion and reaction of planktonic cells, which are able to switch their mode of growth from planktonic to sessile when specific environmental conditions are found. Two systems of non-linear differential equations for the substrate and planktonic cells mass balance within the bulk liquid are also considered. The free boundary evolution is governed by a differential equation that accounts for detachment. The qualitative analysis is performed and a uniqueness and existence result is presented. Furthermore, two special models of biological and engineering interest are discussed numerically. The invasion of Anammox bacteria in a constituted biofilm inhabiting the deammonification units of the wastewater treatment plants is simulated. Numerical simulations are run to evaluate the influence of the colonization process on biofilm structure and activity.

For the quite extensively developed partial differential equations (PDE) backstepping methodology for coupled linear hyperbolic PDEs, we provide a generalization from finite collections of such PDEs, whose states at each location in space are vector-valued, to previously unstudied infinite (continuum) ensembles of such hyperbolic PDEs, whose states are function-valued. The motivation for studying such systems comes from traffic applications (where driver and vehicle characteristics are continuously parametrized), fluid and structural applications, and future applications in population dynamics, including epidemiology. Our design is of an exponentially stabilizing scalar-valued control law for a PDE system in two independent dimensions, one spatial dimension and one ensemble dimension. In the process of generalizing PDE backstepping from finite to infinite collections of PDE systems, we generalize the results for PDE backstepping kernels to the continuously parametrized Goursat-form PDEs that govern such continuously parametrized kernels. The theory is illustrated with a simulation example, which is selected so that the kernels are explicitly solvable, to lend clarity and interpretability to the simulation results.

Solving parabolic and hyperbolic equations by the generalized finite difference method

SUMMARYTo evaluate the computational performance of high‐order elements, a comparison based on operation count is proposed instead of runtime comparisons. More specifically, linear versus high‐order approximations are analyzed for implicit solver under a standard set of hypotheses for the mesh and the solution. Continuous and discontinuous Galerkin methods are considered in two‐dimensional and three‐dimensional domains for simplices and parallelotopes. Moreover, both element‐wise and global operations arising from different Galerkin approaches are studied. The operation count estimates show, that for implicit solvers, high‐order methods are more efficient than linear ones. Copyright © 2013 John Wiley & Sons, Ltd.

The basic error estimate for the discontinuous Galerkin method for hyperbolic equations indicates an O ( h n + 1 2 ) O(h^{n+\frac {1}{2}}) convergence rate for n th n\textrm {th} degree polynomial approximation over a triangular mesh of size h h . However, the optimal O ( h n + 1 ) O(h^{n+1}) rate is frequently seen in practice. Here we extend the class of meshes for which sharpness of the O ( h n + 1 2 ) O(h^{n+\frac {1}{2}}) estimate can be demonstrated, using as an example a problem with a “nonaligned” mesh in which all triangle sides are bounded away from the characteristic direction. The key to realizing h n + 1 2 h^{n+\frac {1}{2}} convergence is a mesh which, to the extent possible, directs the error to lower frequency modes which are approximated, not damped, as h → 0 h\rightarrow 0 .

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.

185 - Hyperbolic Equations: Method of Characteristics