479 research outputs found
Effective Source Term Discretizations for Higher Accuracy Finite Volume Discretization of Parabolic Equations
A finite volume method is applied to develop space-time discretizations for parabolic equations based on an equation error method.A space-time expansion of the local equation error based on flux integral formulation of the equation is first designed using a desiredframework of neighboring quadrature points for the solution and local source terms. The quadrature weights are then determined through aminimization process for the error which constitutes all local compact fluxes about each centroid within the computational domain.In utilizing a local source term distribution to account for diffusive fluxes, the right minimizing quadrature weights and collocationpoints including subgrid points for the source terms may be determined and optimized for higher accuracies as well as robust higher-ordercomputational convergence. The resulting local residuals form a more complete description of the truncation errors which are then utilizedto assess the computational performances of the resulting schemes. The effectiveness of the discretization method is demonstrated by theresults and analysis of the schemes
Higher-Order Accurate Finite Volume Discretization of the Three-Dimensional Poisson Equation Based on An Equation Error Method
Efficient higher-order accurate finite volume schemes are developed for the threedimensional Poisson’s equation based on optimizations of an equation error expansion on local control volumes. A weighted quadrature of local compact fluxes and the flux integral form of the equation are utilized to formulate the local equation error expansions. Efficient quadrature weights for the schemes are then determined through a minimization of the error expansion for higher-order accurate discretizations of the equation. Consequently, the leading numerical viscosity coefficients are more accurately and completely determined to optimize the weight parameters for uniform higher-order convergence suitable for effective numerical modeling of physical phenomena. Effectiveness of the schemes are evaluated through the solution of the associated eigenvalue problem. Numerical results and analysis of the schemes demonstrate the effectiveness of the methodology
Spatially partitioned embedded Runge-Kutta Methods
We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory
Computational Methods and Results for Structured Multiscale Models of Tumor Invasion
We present multiscale models of cancer tumor invasion with components at the
molecular, cellular, and tissue levels. We provide biological justifications
for the model components, present computational results from the model, and
discuss the scientific-computing methodology used to solve the model equations.
The models and methodology presented in this paper form the basis for
developing and treating increasingly complex, mechanistic models of tumor
invasion that will be more predictive and less phenomenological. Because many
of the features of the cancer models, such as taxis, aging and growth, are seen
in other biological systems, the models and methods discussed here also provide
a template for handling a broader range of biological problems
A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations
High-order entropy-stable discontinuous Galerkin methods for the compressible
Euler and Navier-Stokes equations require the positivity of thermodynamic
quantities in order to guarantee their well-posedness. In this work, we
introduce a positivity limiting strategy for entropy-stable discontinuous
Galerkin discretizations constructed by blending high order solutions with a
low order positivity-preserving discretization. The proposed low order
discretization is semi-discretely entropy stable, and the proposed limiting
strategy is positivity preserving for the compressible Euler and Navier-Stokes
equations. Numerical experiments confirm the high order accuracy and robustness
of the proposed strategy
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