The Finite Volume Method

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

This work shows that, by adequately rewriting the one-dimensional convection-diffusion equation in a diffusional form, all the usual numerical procedures, that is, the finite element, finite volume and finite difference procedures, lead to the same and optimal steady-state stencil for numerical solutions. Additionally, a lumped capacitance (mass) matrix first-order time integration technique is presented. The resulting scheme, expressed with a transient, special lumped capacitance matrix (PMGV or prevailing main grid value method), was used to solve the problem of onedimensional transient saturated soil solute transport. A benchmark problem with analytical solution, representing a moving concentration front in a saturated soil was solved by both the diffusional method and by the Two-Step Taylor-Galerkin (TSTG) finite element method. The diffusional (PMGV) method proved itself to be an excellent option for solving advection dominated problems such as soil solute transport problems. The simulation results showed that the explicit PMGV scheme leads to the same accuracy as the TSTG scheme. The TSTG scheme shows numerical dispersion while the diffusional method showed false diffusion. However, the stability region for the PMGV is broader. The PMGV scheme is more effective than the TSTG in problems involving either low Pe numbers or high Pe numbers, if mesh refinement is made.

Real-world groundwater modeling deals with heterogeneous and anisotropic aquifer systems. Numerical models, particularly the finite difference and finite element methods, are extensively used for modeling such a complex aquifer system. The finite difference method is an age-old straightforward approach, whereas the finite element method is more idealistic, relatively new, and has emerged as a popular approach. The unique advantages with the finite element method are (i) an irregular-shaped domain can be discretized into elements with a matching boundary, (ii) a node can be connected with the neighboring nodes in any direction. Thus, the finite element analysis allows water/solute to flow more freely toward the surrounding nodes along nodal links. On the other hand, the mathematical formulation of the finite element technique follows a set of rules step by step. Conventionally, this idealistic approach uses shape functions and integration techniques to develop element matrices, which, in turn, are assembled to form the global matrix. This conventional global matrix was found to be inconsistent with the basic partial differential equation representing the mass balance in a nodal domain as a control volume. As a result, the matrix solution, particularly for nonlinear problems (e.g., vadose zone flow, density dependent flow, and so forth) becomes unstable and computationally burdensome. To overcome the limitations of the conventional finite element method for real-world groundwater modeling, a new approach, called the direct-formulation finite element (DFFE) method, has been developed. In this approach, a two-dimensional model domain is discretized into triangular elements. Element matrices are formed directly based on flow principles and material properties of elements without using shape functions and integration process. The DFFE method is robust. The global matrix is consistent, and the solution is stable and efficient. The results obtained from this approach were found in a good agreement with that obtained from analytical and other widely used models.

Finite difference and finite element methods

The pseudopotential method has grown as a powerful tool for multiphase fluid flow simulations within the lattice Boltzmann method framework. We consider that due to its simplicity and computational efficiency, the pseudopotential method could be explored also inside the framework of more traditional Computational Fluid Dynamic methods such as Finite Difference, Finite Volume or Finite Element methods. Following this idea, in this work we start from the macroscopic equations resulting from the pseudopotential lattice Boltzmann method and discretize it by a simple Finite Difference scheme. This pseudopotential based finite difference method is then tested in different benchmark problems such as a planar interface, a smooth droplet oscillation, and a single droplet evaporation. Excellent results were obtained in all tests. One of the advantages of the proposed method is that mesh refinement is straightforward, and converged solutions can be used as a tool of validation for the lattice Boltzmann method. Results indicate that the pseudopotential method is suitable to be used with standard discretization methods such as Finite Difference and that in future works more robust discretizations can be used to further enhance the pseudopotential method application.

A finite element method (FEM) for solution of the conservative full potential flow equation is presented as a means for improving the computation of transonic flows about complex geometries. The method does not require an orthogonal mesh, thus removing a major constraint in grid generation. A standard Galerkin finite element formulation is used in conjunction with the artificial compressibility method which stabilizes the algorithm in transonic flow and allows the capture of embedded shock waves. Solution comparisons, including one test case which stresses the'capabilities of state-of-the-art finite difference methods, verify the FEM accuracy and geometric capability. I. Introduction W HILE the computation of inviscid two- and threedimensional flowfields by finite difference methods has now become commonplace in industry, one of the most important technological deficiencies is in the area of geometric modeling, particularly for transonic flows. The finite element method (FEM) is presented as a way of alleviating some problems of geometric modeling which limit the effectiveness of current finite difference methods. Certain specific advantages of the FEM, if fully exploited, may improve greatly the computational state-of-the-art. Since the governing equations of inviscid transonic potential flow are nonlinear, the method of superposition (i.e., panel methods) is insufficient to yield a proper solution, even though solutions have been obtained by a method combining surface singularities and field differences.1 Also, because of the nonlinearity, similarity laws do not exist which allow subsonic flows to be related to transonic flows. Field methods must be used, typically finite difference or finite element methods, requiring a grid of points to be defined, both on the body surface arid in the flowfield, on which solutions to the governing equations are approximated. Finite difference method (FDM) computer codes generally

Traditionally reservoir simulation has been accomplished by solution of reservoir flow models using variations of the methods of finite differences and less frequently by finite elements. The accuracy of these methods depends heavily on the size of grid blocks and time steps employed for the numerical solution. Consequently, computational time and memory requirements may be overwhelming in many cases. In the present study, a recently proposed method called the finite analytic method is shown to alleviate the difficulties associated with the conventional methods. The finite analytic method is operationally and principally different than the conventional finite difference and finite element methods in that a general analytical solution of the simulation model is used over mesh elements to obtain a numerical scheme. Therefore, for comparable accuracy, finite analytic method can lead to higher order approximations than those achieved by the finite difference and finite element techniques. Finite analytic schemes are derived from the reservoir flow models and therefore they are naturally problem dependent. This is the reason for the efficiency of the finite analytic method. By this method problems such as numerical diffusion, numerical instability, slow convergence, as well as requirements of large number of time steps and grid blocks are significantly reduced. In this study, the methodology of formulating and calculating with this method is illustrated by means of a typical two-phase immiscible flow problem described by the Buckley-Leverett equation. The results are compared with the analytic solution of Buckley-Leverett problem and solutions obtained by finite difference and finite element methods. The present study demonstrates that the finite analytic method is capable of producing accurate flow behavior in porous media even when relatively large time steps and grid blocks are used.

We couple a node-centered finite volume method to a high order finite difference method to simulate dynamic earthquake ruptures along nonplanar faults in two dimensions. The finite volume method is implemented on an unstructured mesh, providing the ability to handle complex geometries. The geometric complexities are limited to a small portion of the overall domain and elsewhere the high order finite difference method is used, enhancing efficiency. Both the finite volume and finite difference methods are in summation-by-parts form. Interface conditions coupling the numerical solution across physical interfaces like faults, and computational ones between structured and unstructured meshes, are enforced weakly using the simultaneous-approximation-term technique. The fault interface condition, or friction law, provides a nonlinear relation between fields on the two sides of the fault, and allows for the particle velocity field to be discontinuous across it. Stability is proved by deriving energy estimates; stability, accuracy, and efficiency of the hybrid method are confirmed with several computational experiments. The capabilities of the method are demonstrated by simulating an earthquake rupture propagating along the margins of a volcanic plug.

A coupled water and heat transport mode is established based on the Richards equation to study water flow and heat transport in soil during freezing process. Both the finite difference and finite element method are used in the discretization, respectively. Two different computer programs are written and used to simulate an indoor unidirectional frozen test. The freezing depth, freezing rate and temperature variation are compared among lab tests, finite difference calculation simulation and finite element calculation simulation. Result shows that: the finite difference method has a better performance in freezing depth simulation while the finite element method has a better performance in numerical stability in one-dimensional freezing simulation.

B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems

Application of the trajectory image method to solve convection-diffusion equations

Chapter Seven - Fuel Cell Modeling

Bellows compensators are critical components in pipeline systems, designed to absorb thermal expansions, vibrations, and pressure reflections. Ensuring their operational reliability requires accurate prediction of the stress–strain state (SSS) and stability under internal pressure. This study presents a comprehensive mathematical model for analyzing corrugated bellows compensators, formulated as a boundary value problem for a system of partial differential equations (PDEs) within the Kirchhoff–Love shell theory framework. Two numerical approaches are developed and compared: a finite difference method (FDM) applied to a reduced axisymmetric formulation to ordinary differential equations (ODEs) and a finite element method (FEM) for the full variational formulation. The FDM scheme utilizes a second-order implicit symmetric approximation, ensuring stability and efficiency for axisymmetric geometries. The FEM model, implemented in Ansys 2020 R2, provides high fidelity for complex geometries and boundary conditions. Convergence analysis confirms second-order spatial accuracy for both methods. Numerical experiments determine critical pressures based on the von Mises yield criterion and linearized buckling analysis, revealing the influence of geometric parameters (wall thickness, number of convolutions) on failure mechanisms. The results demonstrate that local buckling can occur at lower pressures than that of global buckling for thin-walled bellows with multiple convolutions, which is critical for structural reliability assessment. The proposed combined approach (FDM for rapid preliminary design and FEM for final verification) offers a robust and efficient methodology for bellows design, enhancing reliability and reducing development time. The work highlights the importance of integrating rigorous PDE-based modeling with modern numerical techniques for solving complex engineering problems with a focus on structural integrity and long-term performance.

In this chapter we discuss the prerequisites that are necessary to understand how problems in the natural sciences are formulated using (differential) equations and how to solve them numerically. For this, we discuss the basics of ordinary and partial differential equations in Sect. 4.1. We learn about the Lipschitz condition for ordinary differential equations and about initial and boundary conditions with partial differential equations, where we distinguish between three basic types of equations. A discussion of plane and spherical wave solutions of partial differential equations leads us to the study of numerical procedures in Sect. 4.2. Here, we discuss mesh-based and mesh-free methods, finite difference, finite volume, and finite element methods. We also get to know the most important algorithm used for numerical integration in molecular dynamics methods – the velocity Verlet algorithm. Finally, in Sect. 4.3 some tricks of the trade and important principles in software design and engineering which are not found in other books on computational science are presented. The chapter ends with a couple of problems for the interested reader.

A Boussinesq-type wave model is developed in this paper to simulate the interaction of coastal waves with bottom-mounted porous structures. The governing equations are rewritten in the conservative form to facilitate the use of hybrid finite volume (FV) and finite difference (FD) method. Higher-order slope terms are also inserted into the equations to account for rapidly varying bathymetry. The convective flux is approximated using the FV method, while the remaining terms are discretized using the FD method in a uniform rectangle grid system. The time integration is implemented using the third order Runge–Kutta method with an adaptive time step. A single GPU parallel computation is also implemented to save computation costs. The numerical model is validated against a series of experimental datasets, including data acquired in a new laboratory experiment. The predictions are in overall agreement with the measurements, proving that the model is capable of handling wave interaction with porous structures in the coastal region for a wide range of scenarios.