Multigrid stabilization of time integrators for nonlinear parabolic PDEs

: To illustrate Brandt's multigrid method this report considers the Helmholtz equation in cases where the frequency is close to an eigenfrequency. In a second part the eigenvalue problem is discussed. The basic idea for the treatment of the inhomogeneous problem is contained in Brandt's publications. Difficulties are encountered because the coarse grid eigenvalues differ from those of the finer grid. As a consequence, approximations to a solution in the coarse grid may differ widely from the corresponding approximation in a finer grid, if the frequency defined by the inhomogeneous problem is close to one of the eigenfrequencies. This may be detrimental to the aim of the coarse grid step; namely to remove from an approximation in a fine grid long waves which cause divergence of fine grid iterations. This is remedied by eliminating long wave contributions to the residual directly in the next finer grid. The success of this procedure still depends upon the choice of the subspace from which the long wave corrections in the finer grid are taken. Even this modified method fails if the frequency pertaining to the inhomogeneous term is too close to an eigenvalue, unless the subspace contains the pertinent eigenfunction.

Flow fields in a centrifugal fan for an indoor unit of an air-conditioner were calculated with finite element method-based large eddy simulation (LES) with the aim of predicting fan performance and aerodynamic noise in this study. The numerical simulation code employed throughout the LES was called FrontFlow/blue (FFB). We compared 10M grid [coarse grid] and 60M grid [fine grid] calculation results for investigation of influence of grid resolution. In the fine grid, the number of grid elements in blade-to-blade direction, and of region between the shroud and the bell mouth increased in particular. By calculating with the fine grid, calculated distributions of absolute velocities at blade exit reasonably agreed with experimental results. Because of this, maximum absolute velocity by fine grid near hub decreased as compared to those by coarse grid. Calculated sound pressure level by fine grid was therefore smaller than that by coarse grid, and the overestimation of sound pressure was suppressed by calculating with fine grid. This decrease of the absolute velocity was a first factor for the improvement of calculation accuracy. Moreover, number of captured streaks on the blade, hub, and shroud surfaces by fine grid increased as compared to those by coarse grid. As a result, size of streak by fine grid became smaller than that by coarse grid. Static pressure fluctuations by fine grid on the blade, hub, and shroud surfaces therefore reduced as compared to those by coarse grid. Aerodynamic noise was related to static pressure fluctuations according to Curle’s equation. This reduction of static pressure fluctuations was therefore a second factor for improvement of calculation accuracy.

Running a fine grid model with 107 - 109 of cells is possible using a supercomputer with 103 - 106 of CPUs but may not be always cost-effective. The most cost-effective way is to use a coarse grid model that is much smaller but with static/dynamic profiles very close to the fine grid model. This paper proposes a new layer optimization and upscaling method with the aim for creating a consistent coarse grid model. Unlike the industry's existing layer optimization and upscaling methods, the proposed method performs layer optimization and upscaling fully integrated with the Lorenz coefficient and curves (LCC). Coarse grid layers and their permeabilities are created by minimizing the difference between fine and coarse grid LCCs. The process consists of static and dynamic optimizations. The former is measured by LCC while the latter by pressure, GOR, and water-cut. A new LCC-based permeability upscaling method is developed to preserve the fine grid multiphase flow behaviors. A satisfactory coarse grid model is achieved when both static and dynamic criteria are met. The proposed method has been successfully applied to a giant carbonate oil field in the Caspian Sea that consists of a matrix dominated platform and a fracture/karst dominated rim. Due to the field's complex geology and high H2S content (15%), a dual porosity, dual permeability compositional model has been created to model compositional sour crude flow within and between the matrix and fracture/karst features. The reservoir drive mechanisms are fluid expansion, miscible gas injection and aquifer drive. The reservoir is undersaturated and has an abnormally high initial reservoir pressure. The fine-grid static model contains 104 million cells (370×225×625×2) and the optimized upscaled coarse-grid dynamic model has 8.3 million cells (370×225×50×2). The upscaled model can be run efficiently on the company's existing HPC infrastructure with a maximum of 64 CPUs. Excellent matches of the Lorenz coefficient maps for reservoir total/zones and Lorenz curves at all wells between the fine and coarse grid models have been achieved. Matches on the dynamic variables, e.g., pressure, gas breakthrough time, and GOR growth, in all producers are within the defined acceptable tolerances. The high quality of the static and dynamic matches between the coarse- and fine-grid models confirms that the reservoir properties of the coarse-grid model is very close to the fine-grid model and can be used a base model for history matching and uncertainty analysis.

Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results.

Fast and robust sixth-order multigrid computation for the three-dimensional convection–diffusion equation

We propose two improvements to the Reitzinger and Schoberl algebraic multigrid (AMG) method for solving the eddy current approximations to Maxwell's equations. The main focus in the Reitzinger/Schoberl method is to maintain null space properties of the weak $\nabla \times \nabla \times$ operator on coarse grids. While these null space properties are critical, they are not enough to guarantee h-independent convergence of the overall multigrid method. We illustrate how the Reitzinger/Schoberl AMG method loses h-independence due to the somewhat limited approximation property of the grid transfer operators. We present two improvements to these operators that not only maintain the important null space properties on coarse grids but also yield significantly improved multigrid convergence rates. The first improvement is based on smoothing the Reitzinger/Schoberl grid transfer operators. The second improvement is obtained by using higher order nodal interpolation to derive the corresponding AMG interpolation operators. While not completely h-independent, the resulting AMG/CG method demonstrates improved convergence behavior while maintaining low operator complexity.

Boundedness of global solutions of nonlinear parabolic equations

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Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions

Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

Nonlinear Parabolic Equations with Measure Data

Variational iteration method is a semi analytic solution used to solve the parabolic differential equation both of homogen or nonhomogen. In the process of determining an approximation solution, this method did not use a linearization and a small pertubation. In this paper, the variational iteration method is implemented in the parabolic differential equation in the form of ut = uxx + f(u) + g(x, t) with appropriate intial condition. Furthermore, some examples of special parabolic differential equations are given to test the reliability and convergence of the method. Based on the result of study shows that the variational iteration method is able to solve the parabolic differential equation with a good accuration.

Consider the first initial-boundary value problem $$u = 0\quad on\quad \partial \Omega \times \left( {0,\infty } \right)$$ (1.1) , $$u = 0\quad on\quad \partial \Omega \times \left( {0,\infty } \right)$$ (1.2) , $$u\left( {x,0} \right) = \phi (x)\quad on\quad \Omega$$ (1.3) When Ω is a bounded domain in ℝ N with C 2 boundary, $$Lu = \sum\limits_{i,j = 1}^N {\frac{\partial }{{\partial {x_i}}}} \left( {{a_{ij}}\left( x \right){u_{{x_j}}}} \right) + \sum\limits_{i = 1}^N {{b_i}\left( x \right)} {u_i} + c\left( x \right)u$$ (aij) is uniformly positive definite matrix, the coefficients of L are sufficiently smooth in (math), say in C 1,ϕ is Holder continuous in Ω;, and F(u, w) is Lipschitz continuous in ℝ N +1.

We reduce the construction of a weak solution of the Cauchy problem for a quasilinear parabolic equation to the construction of a solution to a stochastic problem. Namely, we construct a diffusion process that allows us to obtain a probabilistic representation of a weak (in distributional sense) solution to the Cauchy problem for a nonlinear PDE.

Large solutions for nonlinear parabolic equations without absorption terms