Linear Convection Equation Research Articles

A fourth-order compact finite volume method on unstructured grids for simulation of two-dimensional incompressible flow

This paper introduces a novel and efficient compact reconstruction procedure for high-order finite volume methods applied to unstructured grids. In this procedure, we establish a set of constitutive relations that ensure the continuity of the reconstruction polynomial and its normal derivatives between adjacent elements at control points. The paper delves into the details of the fourth-order compact reconstruction method specifically designed for two-dimensional triangular grids. This method can be considered an extension of the one-dimensional compact scheme and cubic spline interpolation methods to two-dimensional triangular unstructured grids. In the cubic polynomial reconstruction, a two-dimensional cubic polynomial is reconstructed using the cell averages of elements in the standard stencil and the function values at control points. The determination of function values at control points is achieved by adhering to the constraint of normal derivative continuity. This reconstruction is solved using a relaxation iteration method and has been verified to be solvable for triangular grids. Compared to other fourth-order implicit compact polynomial reconstructions, our method requires solving a smaller number of unknowns, which means less computational cost in reconstruction. By applying this method to solve the Poisson equation, the linear convection equation, and incompressible flow benchmarks, it demonstrates that the proposed method exhibits the expected high-order accuracy and performs well in incompressible flow problems.

Evaluating Numerical Stability in High-Accuracy Simulations: A Comparative Study of Time Discretization Methods for the Linear Convection Equation

Evaluating Numerical Stability in High-Accuracy Simulations: A Comparative Study of Time Discretization Methods for the Linear Convection Equation

Quantum simulation for partial differential equations with physical boundary or interface conditions

This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretization of such problems does not necessarily yield Hamiltonian dynamics and even alters the Hamiltonian structure of the dynamics when boundary and interface conditions are included. This seemingly intractable issue can be resolved by using a recently introduced Schrödingerisation method [1,2] – it converts any linear PDEs and ODEs with non-Hermitian dynamics to a system of Schrödinger equations, via the so-called warped phase transformation that maps the equation into one higher dimension. We implement this method for several typical problems, including the linear convection equation with inflow boundary conditions and the heat equation with Dirichlet and Neumann boundary conditions. For interface problems we study the (parabolic) Stefan problem, linear convection, and linear Liouville equations with discontinuous and even measure-valued coefficients. We perform numerical experiments to demonstrate the validity of this approach, which helps to bridge the gap between available quantum algorithms and computational models for classical and quantum dynamics with boundary and interface conditions.

Inclusion of anisotropy in understanding rock deformation and inter-well fracture growth in layered formation through CZM based XFEM

In layered formations such as the Wufeng-Longmaxi marine shale, mechanical rock properties may be uniform horizontally within a layer, but vary vertically, and from layer to layer. In this study, we implement Newton-Raphson method within a rigorous two-dimensional scheme for computing fluid-solid coupling at every time increment in an anisotropic rock frame. The investigated domain consists of intercalations of soft and hard rock layers with a consideration of a cluster of three horizontal wells. The linear convection equation describing the fluid flow is discretized using upwind scheme whose integration points are determined by the intersection between the crack geometry and the underlying discretization of the extended finite element method (XFEM). A normalized slurry concentration averaged over the fracture width is used to establish a basis for analysing fluid viscosity effects with subsequent XFEM computation indicating that an increased slurry concentration leads to increasing fracture aperture. We further conduct a thorough investigation of an inter-well fracture development with the analyses revealing that the fractures kink at the interfaces of the intercalations of soft and hard layers. It is also observed that, when the elastic contrast between the soft and hard strata decreases, fracture kinking reduces and the inter-well fracture interference mechanism switches to linking or coalescence. Before the linking, the fluid pressure fields inside the separate fractures are independent, whereas an equilibrated pressure field develops inside the inter-linked fracture system. Oriented initial perforations show no link-growth of fractures. However, we observe that the highest inclination angle, which is 75° in our case, yields the highest treatment pressure of 62.8 MPa, whereas the lowest inclination angle of 15° yields the lowest treatment pressure of 56.3 MPa. We validate our results through comparison with a triaxial fracturing cell experiment, and Kristianovich-Geertsma-de Klerk (KGD) fracture problem. In all cases, the findings agree with the numerical results.

A New Third-Order Finite Difference WENO Scheme to Improve Convergence Rate at Critical Points

In this work, a new, improved third-order finite difference weighted essentially non-oscillatory scheme is presented for one- and two-dimensional hyperbolic conservation laws and associated problems. The parameter p which is regulate dissipation is introduced in the nonlinear weights in the framework of the conventional WENO-Z scheme, and the higher-order global smoothness indicator is obtained by the idea of Wang [Wang, Y. H., Y. L. Du, K. L. Zhao and L. Yuan. 2020. ‘A Low-dissipation Third-order Weighted Essentially Nonoscillatory Scheme with a New Reference Smoothness Indicator’. International Journal for Numerical Methods in Fluids. 92 (9): 1212–1234.], the sufficient condition of nonlinear weights is proved by using Taylor expansions. Finally, the value range of parameter p is obtained. The proposed scheme is verified to achieve the optimal order near critical points by linear convection equations with different initial values, and the high-resolution characteristic of the present scheme is proved on a variety of one- and two- dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with the other third-order WENO schemes.

Numerical Simulation of Singularity Propagation Modeled by Linear Convection Equations with Spatially Heterogeneous Nonlocal Interactions

We study the propagation of singularities in solutions of linear convection equations with spatially heterogeneous nonlocal interactions. A spatially varying nonlocal horizon parameter is adopted in the model, which measures the range of nonlocal interactions. Via heterogeneous localization, this can lead to the seamless coupling of the local and nonlocal models. We are interested in understanding the impact on singularity propagation due to the heterogeneities of the nonlocal horizon and the local and nonlocal transition. We first analytically derive equations to characterize the propagation of different types of singularities for various forms of nonlocal horizon parameters in the nonlocal regime. We then use asymptotically compatible schemes to discretize the equations and carry out numerical simulations to illustrate the propagation patterns in different scenarios.

Analysis of Pseudo-spectral Methods Used for Numerical Simulations of Turbulence

Events in turbulent flows computed by direct numerical simulation (DNS) are often calibrated with properties based on homogeneous isotropic turbulence, advanced by Kolmogorov, and given in Turbulence, U. Frisch, Cambridge Univ. Press, UK (1995). However, these computational procedures are not calibrated using numerical analyses in order to assess their strengths and weaknesses for DNS. This is with the exception in "A critical assessment of simulations for transitional and turbulence flows- Sengupta, T.K., In Proc. of IUTAM Symp. on Advances in Computation, Modeling and Control of Transitional and Turbulent Flows, 491-532, (eds. Sengupta, Lele, Sreenivasan, Davidson), WSPC, Singapore (2016)", where such a calibration has been advocated for numerical schemes using global spectral analysis (GSA) for the convection equation. In recent times, due to growing computing power, simulations have been reported using pseudo-spectral methods, with spatial discretization performed in Fourier spectral space and time-integration by multi-stage Runge-Kutta (RK) methods. Here, we perform GSA of Fourier spectral methods for the first time with RK2 and other multistage Runge-Kutta methods using the model linear convection and convection-diffusion equations. With the help of GSA, various sources of numerical errors are quantified. The major limitations of the RK2-Fourier spectral method are demonstrated for DNS and alternate choices are presented. We specify optimal parameters to achieve the best possible accuracy for simulations. There is a one-to-one correspondence of numerical solutions obtained by linear convection-diffusion equation and nonlinear Navier-Stokes equation with respect to numerical parameters. This enables us to investigate the capabilities of the numerical methods for DNS.

A new WENO weak Galerkin finite element method for time dependent hyperbolic equations

In this paper, we develop a new WENO weak Galerkin finite element scheme for solving the time dependent hyperbolic equations. The upwind-type stabilizer is imposed to enforce the flux direction in the scheme. For the linear convection equations, we analyze the L2-stability and error estimate for L2-norm. We also investigate a simple limiter using weighted essentially non-oscillatory (WENO) methodology for obtaining a robust procedure to achieve high order accuracy and capture the sharp, non-oscillatory shock transitions. The approach applies for linear convection equations and Burgers equations. Finally, numerical examples are presented for validating the theoretical conclusions.

Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension

Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension

Superconvergent Functional Estimates from Tensor-Product Generalized Summation-by-Parts Discretizations in Curvilinear Coordinates

We investigate superconvergent functional estimates in curvilinear coordinates for diagonal-norm tensor-product generalized summation-by-parts operators. We show that interpolation/extrapolation operators of degree greater than or equal to 2p are required to preserve at least 2p quadrature accuracy and functional superconvergence in curvilinear coordinates when: (1) the Jacobian of the coordinate transformation is approximated by the same generalized summation-by-parts operator that is used to approximate the flux terms and (2) the degree of the generalized summation-by-parts operator is lower than the degree of the polynomial used to represent the geometry of interest. Legendre–Gauss–Lobatto and Legendre–Gauss element-type operators are considered. When the aforementioned condition (2) is violated for the Legendre–Gauss operators, there is an even–odd quadrature convergence pattern that is explained by the cancellation of the leading truncation error terms for the interpolation/extrapolation operators that correspond to the odd-degree Legendre–Gauss operators. The theory developed is confirmed through numerical examples with a steady one-dimensional problem and the unsteady two-dimensional linear convection equation.

CFD Julia: A Learning Module Structuring an Introductory Course on Computational Fluid Dynamics

CFD Julia is a programming module developed for senior undergraduate or graduate-level coursework which teaches the foundations of computational fluid dynamics (CFD). The module comprises several programs written in general-purpose programming language Julia designed for high-performance numerical analysis and computational science. The paper explains various concepts related to spatial and temporal discretization, explicit and implicit numerical schemes, multi-step numerical schemes, higher-order shock-capturing numerical methods, and iterative solvers in CFD. These concepts are illustrated using the linear convection equation, the inviscid Burgers equation, and the two-dimensional Poisson equation. The paper covers finite difference implementation for equations in both conservative and non-conservative form. The paper also includes the development of one-dimensional solver for Euler equations and demonstrate it for the Sod shock tube problem. We show the application of finite difference schemes for developing two-dimensional incompressible Navier-Stokes solvers with different boundary conditions applied to the lid-driven cavity and vortex-merger problems. At the end of this paper, we develop hybrid Arakawa-spectral solver and pseudo-spectral solver for two-dimensional incompressible Navier-Stokes equations. Additionally, we compare the computational performance of these minimalist fashion Navier-Stokes solvers written in Julia and Python.

A level set method for shape optimization in semilinear elliptic problems

We develop a finite-element based level set method for numerically solve shape optimization problems constrained by semilinear elliptic problems. By combining the shape sensitivity analysis and level set method, a gradient descent algorithm is proposed to solve the model problem. Different from solving the nonlinear Hamilton–Jacobi equations with finite differences in traditional level set methods, we solve the linear convection equation and reinitialization equation using the characteristic Galerkin finite element method. The methodology can handle topology as well as shape changes in both regular and irregular design regions. Numerical results are presented to demonstrate the effectiveness of our algorithm as well as to verify symmetry preserving and breaking properties of optimal subdomains.

Totally Volume Integral of Fluxes for Discontinuous Galerkin Method (TVI-DG) I-Unsteady Scalar One Dimensional Conservation Laws

The volume integral of Riemann flux in the discontinuous Galerkin (DG) method is introduced in this paper. The boundaries integrals of the fluxes (Riemann flux) are transformed into volume integral. The new family of DG method is accomplished by applying divergence theorem to the boundaries integrals of the flux. Therefore, the (DG) method is independent of the boundaries integrals of fluxes (Riemann flux) at the cell (element) boundaries as in classical (DG) methods. The modified streamline upwind Petrov-Galerkin method is used to capture the oscillation of unphysical flow for shocked flow problems. The numerical results of applying totally volume integral discontinuous Galerkin method (TVI-DG) are presented to unsteady scalar hyperbolic equations (linear convection equation, inviscid Burger's equation and Buckley-Leverett equation) for one dimensional case. The numerical finding of this scheme is very accurate as compared with other high order schemes as the weighted compact finite difference method WCOMP.

A high-order finite volume method for solving one-dimensional convection and diffusion equations

ABSTRACTA spatially high-order finite volume method for solving convection and diffusion equations is developed and tested in this work. The method performs a high-order piecewise polynomial reconstruction of the local flow field based on the relationship between Taylor’s series expansion and the volume-averaged flow quantities. A 5 × 5 matrix inversion for each cell is done to compute the cell-center variables and derivatives up to fourth order. While a fixed symmetric grid stencil is maintained in smooth flow regions, a detector for large change in linear data slopes is developed to trigger the use of ENO stencil around flow discontinuities. Regular time integration scheme such as the four-stage Runge–Kutta method or the Euler implicit method is used for time integration. The present finite volume method is shown to be spatially fifth-order accurate for the linear convection equation, fourth-order accurate for the linear diffusion equation, and fourth-order accurate for the linear convection–diffusion equation. The shocks captured in solving the inviscid Burger’s equation are sharp and oscillation free. For the system of Euler equations, a characteristic limiter is further developed to limit the growth of total variation of the solution. Test examples solving shock-tube problems and the interactions of two blast waves show that various flow discontinuities are captured sharply without spurious oscillations.

Corner-corrected diagonal-norm summation-by-parts operators for the first derivative with increased order of accuracy

Combined with simultaneous approximation terms, summation-by-parts (SBP) operators offer a versatile and efficient methodology that leads to consistent, conservative, and provably stable discretizations. However, diagonal-norm operators with a repeating interior-point operator that have thus far been constructed suffer from a loss of accuracy. While on the interior, these operators are of degree 2p, at a number of nodes near the boundaries, they are of degree p, and therefore of global degree p — meaning the highest degree monomial for which the operators are exact at all nodes. This implies that for hyperbolic problems and operators of degree greater than unity they lead to solutions with a global order of accuracy lower than the degree of the interior-point operator. In this paper, we develop a procedure to construct diagonal-norm first-derivative SBP operators that are of degree 2p at all nodes and therefore can lead to solutions of hyperbolic problems of order 2p+1. This is accomplished by adding nonzero entries in the upper–right and lower–left corners of SBP operator matrices with a repeating interior-point operator. This modification necessitates treating these new operators as elements, where mesh refinement is accomplished by increasing the number of elements in the mesh rather than increasing the number of nodes. The significant improvements in accuracy of this new family, for the same repeating interior-point operator, are demonstrated in the context of the linear convection equation.

A High-Order Piecewise Polynomial Reconstruction for Finite-Volume Methods Solving Convection and Diffusion Equations

In finite-volume methods for fluid flows, the average of field variables over local mesh cells are the unknowns that are integrated in time based on the integral conservation laws. In order to compute the cell-face fluxes, nodal variables and derivative values at cell faces are needed, which ultimately determine the accuracy of the finite-volume method. In this work, a piecewise fourth-order polynomial reconstruction model based on volume averages is developed for smoothly varying flow fields over finite-volume cells. The obtained polynomial is used to extrapolate the cell-face variables and derivative values with high-order accuracy. The extrapolated cell-face quantities are used directly to compute the convection or diffusion integrals to construct high-order finite-volume methods. Some one-dimensional examples are shown to demonstrate the fifth-order accuracy of the proposed approach when solving the linear convection equation, and the fourth-order solution accuracy when solving the linear diffusion equation.

Stability of natural ventilation mode after main fan stoppage

Results of seasonal experimental investigations of natural draft in the potash mines after main fan stoppage are presented and analyzed. Underground mines with equal top elevations are considered. Three modes of natural ventilation depending on the season are determined: flow-through, local and convective ventilation modes. The mathematical model of unsteady heat exchange between mine air, shaft support and rock mass is proposed. This model allows prediction of natural draft intensity for flow-through ventilation mode. Also using system of linear convection equations we formulate natural ventilation stability criterion in case of convective natural ventilation mode for mine shafts after main fan stoppage. Formulated criterion based on Rayleigh number allows prediction of convective ventilation mode formation in shafts.

Lie group analysis method for two classes of fractional partial differential equations

In this paper we deal with two classes of fractional partial differential equation: n order linear fractional partial differential equation and nonlinear fractional reaction diffusion convection equation, by using the Lie group analysis method. The infinitesimal generators general formula of n order linear fractional partial differential equation is obtained. For nonlinear fractional reaction diffusion convection equation, the properties of their infinitesimal generators are considered. The four special cases are exhaustively investigated respectively. At the same time some examples of the corresponding case are also given. So it is very convenient to solve the infinitesimal generator of some fractional partial differential equation.

L 1-error estimates on the immersed interface upwind scheme for linear convection equations with piecewise constant coefficients: A simple proof

A linear convection equation with discontinuous coefficients arises in wave propagation through interfaces. An interface condition is needed at the interface to select a unique solution. An upwind scheme that builds this interface condition into its numerical flux is called the immersed interface upwind scheme. An l1-error estimate of such a scheme was first established by Wen et al. (2008). In this paper, we provide a simple analysis on the l1-error estimate. The main idea is to formulate the solution to the underline initial-value problem into the sum of solutions to two convection equations with constant coefficients, which can then be estimated using classical methods for the initial or boundary value problems.

High Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation

High Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation