A Guide to Numerical Methods for Transport Equations

1 A Guide to Numerical Methodsfor Transport EquationsDmitri Kuzmin2010 Contents1 Getting Started.. Introduction to Flow Simulation .. Mathematics of Transport Phenomena .. Principles .. and Diffusive Fluxes .. Generic Transport Equation .. and Boundary Conditions .. Residual Formulation .. Taxonomy of Reduced Models .. Transport Equations .. Transport Equations .. Transport Equations .. of Model Problems .. Space Discretization Techniques .. Meshes .. Problem .. Difference Methods .. Volume Methods .. Element Methods .

2 Systems of Algebraic Equations .. Techniques .. vs. Iterative Solvers .. vs. Implicit Schemes .. Fundamental Design Principles .. Analysis .. Constraints .. Basic Rules .. Scope of This Book .. 38vviContents2 Finite Element Approximations.. Discretization on Unstructured Meshes .. Finite Element Formulation .. of Discrete Operators .. and Mass Lumping .. Gradient Recovery .. of Nonlinear Fluxes .. Flux Decomposition .. to Finite Volumes .. Data Structures .. Row Storage.

3 Stabilization of Convective Terms .. Upwinding .. Diffusion .. Upwinding .. Methods .. Methods .. Capturing .. Penalty Methods .. Dissipation .. Discontinuous Galerkin Methods .. DG Formulation .. Basis Functions .. Barth-Jespersen Limiter .. Vertex-Based Limiter .. Higher-Order Terms .. Summary .. 903 Maximum Principles.. Properties of Linear Transport Models .. Laplace Operator .. of Elliptic Type .. of Hyperbolic Type .. of Parabolic Type .. Perturbed Problems.

4 Matrix Analysis for Steady Problems .. Discrete Problem .. and Monotonicity .. Maximum Principles .. Mesh Properties .. Matrix Analysis for Unsteady Problems .. DMP Constraints .. Discrete DMP Constraints .. Time-Stepping Methods .. Summary .. 124 Contentsvii4 Algebraic Flux Correction.. Nonlinear High-Resolution Schemes .. Philosophy and Tools .. Diffusion Operators .. Flux Decomposition .. Antidiffusive Correction .. Generic Limiting Strategy .. of Algorithmic Steps .. Solution of Nonlinear Systems .. Approximations.

5 Correction Schemes .. and Smoothing .. Solvers .. vs. Convergence .. Steady Transport Problems .. Flux Correction .. to TVD Limiters .. Slope Limiting .. of Local Stencils .. Dissipation .. Examples .. Unsteady Transport Problems .. FEM-FCT Schemes .. s Limiter Revisited .. Linearization Techniques .. Algorithms .. Time Integrators .. Examples .. Limiting for Diffusion Operators .. Galerkin Discretization .. Splitting .. Slope Limiter .. of Nonlinearities .. Examples .. Summary .. 1955 Error Estimates and Adaptivity.

6 Introduction .. Galerkin Weak Form .. Global Error Estimates .. Local Error Estimates .. Numerical Experiments .. Summary .. 202 References.. 205 Chapter 1 Getting StartedIn this chapter, we start with a brief introduction to Numerical simulation of transportphenomena. We consider mathematical models that express certain conservationprinciples and consist of convection-diffusion-reactionequations written in integral,differential, or weak form. In particular, we discuss the qualitative properties ofexact solutions to model problems of elliptic, hyperbolic,and parabolic type.

7 Next,we review the basic steps involved in the design of numericalapproximations andthe main criteria that a reliable algorithm should chapter concludes withan outline of the rationale behind the scope and structure ofthe present Introduction to Flow SimulationFluid dynamics and Transport phenomena, such as heat and mass transfer, play avitally important role in human life. Gases and liquids surround us, flow inside ourbodies, and have a profound influence on the environment in which we live. Fluidflows produce winds, rains, floods, and hurricanes. Convection and diffusion are re-sponsible for temperature fluctuations and Transport of pollutants in air, water or ability to understand, predict, and control Transport phenomena is essential formany industrial applications, such as aerodynamic shape design, oil recovery froman underground reservoir, or multiphase/multicomponent flows in furnaces, heat ex-changers, and chemical reactors.

8 This ability offers substantial economic benefitsand contributes to human well-being. Heating, air conditioning, and weather fore-cast have become an integral part of our everyday life. We take such things forgranted and hardly ever think about the physics and mathematics behind traditional approach to investigation of a physical process is based on ob-servations, experiments, and measurements. The amount of information that canbe obtained in this way is usually very limited and subject tomeasurement , experiments are only possible when a small-scalemodel or the actualequipment has already been built.

9 An experimental investigation may be very time-consuming, dangerous, prohibitively expensive, or impossible for another Getting StartedAlternatively, an analytical or computational study can beperformed on the basisof a suitable mathematical model. As a rule, such a model consists of several differ-ential and/or algebraic Equations which make it possible topredict how the quanti-ties of interest evolve and interact with one another. A drawback to this approach isthe fact that complex physical phenomena give rise to complex mathematical equa-tions that cannot be solved analytically, , using paperand most detailed models of fluid flow are based on first principles , such as theconservation of mass, momentum, and energy.

10 Mathematical Equations that embodythese fundamental principles have been known for a very longtime but used to bepractically worthless until Numerical Methods and digitalcomputers were second half of the twentieth century has witnessed the advent ofComputationalFluid Dynamics(CFD), a new branch of applied mathematics that deals with numer-ical simulation of fluid flows. Nowadays, computer codes based on CFD models areused routinely to predict a variety of increasingly complexflow quality of simulation results depends on the choice of the model and on theaccuracy of the Numerical method.