Related papers: Positivity-Preserving Entropy-Based Adaptive Filte…
Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical…
The main result in this paper is a provably entropy stable shock capturing approach for the high order entropy stable DGSEM based on a hybrid blending with a subcell low order variant. Since it is possible to rewrite a high order SBP…
This paper extends a new class of positivity-preserving, entropy stable spectral collocation schemes developed for the one-dimensional compressible Navier-Stokes equations in [1,2] to three spatial dimensions. The new high-order schemes are…
We combine Patankar-type methods with suitable relaxation procedures that are capable of ensuring correct dissipation or conservation of functionals such as entropy or energy while producing unconditionally positive and conservative…
We present a novel technique for the imposition of non-linear entropy conservative and entropy stable solid wall boundary conditions for the compressible Navier-Stokes equations in the presence of an adiabatic wall, or a wall with a…
The discontinuous Galerkin (DG) finite element method when applied to hyperbolic conservation laws requires the use of shock-capturing limiters in order to suppress unphysical oscillations near large solution gradients. In this work we…
This paper presents a novel structure-preserving scheme for Euler equations, focusing on the numerical conservation of entropy and kinetic energy. Explicit flux functions engineered to conserve entropy are introduced within the…
We present a new class of efficient and robust discontinuous spectral-element methods of arbitrary order for nonlinear hyperbolic systems of conservation laws on curved triangular and tetrahedral unstructured grids. Such discretizations…
In this article we describe the applications of the relative entropy framework. In particular uniqueness of an entropy solution is proven for a scalar conservation law, using the notion of measure-valued entropy solutions. Further we survey…
In this work, we develop a new compatible finite element formulation of the thermal shallow water equations that conserves energy and mathematical entropies given by buoyancy-related quadratic tracer variances. Our approach relies on…
In this paper, a new strategy for a sub-element based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low to high order…
In this work we develop and analyze an adaptive finite element method for efficiently solving electrical impedance tomography -- a severely ill-posed nonlinear inverse problem for recovering the conductivity from boundary voltage…
We present an adaptive variational procedure for unstructured meshes to capture fluid-fluid interfaces in two-phase flows. The two phases are modeled by the phase-field finite element formulation, which involves the conservative Allen-Cahn…
In this paper, we present a shock capturing discontinuous Galerkin (SC-DG) method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time…
Numerical models of weather and climate critically depend on long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical…
This article concerns the development of a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for the multicomponent, chemically reacting, compressible Navier-Stokes equations with complex…
In this paper we present a family of high order cut finite element methods with bound preserving properties for hyperbolic conservation laws in one space dimension. The methods are based on the discontinuous Galerkin framework and use a…
Stability is an important aspect of numerical methods for hyperbolic conservation laws and has received much interest. However, continuity in time is often assumed and only semidiscrete stability is studied. Thus, it is interesting to…
An exact discontinuity capturing central solver developed recently, named MOVERS (Method of Optimal Viscosity for Enhanced Resolution of Shocks, J Computat Phys 2009;228:770-798), is analyzed and improved further to make it entropy stable.…
The upwind conservation element and solution element (CESE) scheme is an alternative discontinuity-capturing numerical approach to solving hyperbolic conservation laws. To evaluate the numerical properties of this spatiotemporal coupled…