Nonlinear Degenerate Parabolic Equations Research Articles

On the Structure of Entropy Solutions to the Riemann Problem for a Degenerate Nonlinear Parabolic Equation

On the Structure of Entropy Solutions to the Riemann Problem for a Degenerate Nonlinear Parabolic Equation

Nonlinear Degenerate Parabolic Equations with a Singular Nonlinearity

Nonlinear Degenerate Parabolic Equations with a Singular Nonlinearity

Interior Holder Regularity and Infinite Time Extinction to a doubly Nonlinear Degenerate Parabolic Equations of Signed Solutions

We study doubly nonlinear parabolic equation with sign changing solutions. We established the Holder regularity and the infinite time extinction of it within a parabolic domain. GANIT J. Bangladesh Math. Soc. 43.1 (2023) 75- 91

On the structure of weak solutions of the Riemann problem for a degenerate nonlinear diffusion equation

An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coe cient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a nite number of variables. A similar result is true for Stefan’s problem. In the limit, when the number of phases tends to in nity, there arises a variational formulation of self-similar solutions to the equation with an arbitrary nonnegative diffusion function.

Entropy dissipative higher order accurate positivity preserving time-implicit discretizations for nonlinear degenerate parabolic equations

We develop entropy dissipative higher order accurate local discontinuous Galerkin (LDG) discretizations coupled with Diagonally Implicit Runge–Kutta (DIRK) methods for nonlinear degenerate parabolic equations with a gradient flow structure. Using the simple alternating numerical flux, we construct DIRK-LDG discretizations that combine the advantages of higher order accuracy, entropy dissipation and proper long-time behavior. We theoretically prove the entropy dissipation of the implicit Euler-LDG discretization without any time-step restrictions when no positivity constraint is imposed. Next, in order to ensure the positivity of the numerical solution, we use the Karush–Kuhn–Tucker (KKT) limiter, which achieves a positive solution by coupling the positivity preserving KKT conditions with higher order accurate DIRK-LDG discretizations using Lagrange multipliers. In addition, mass conservation of the positivity-limited solution is ensured by imposing a mass conservation equality constraint to the KKT equations. Under a time step restriction, the unique solvability and entropy dissipation for implicit first order accurate in time, but higher order accurate in space, positivity-preserving LDG discretizations with periodic boundary conditions are proved, which provide a first theoretical analysis of the KKT limiter. Finally, numerical results demonstrate the higher order accuracy and entropy dissipation of the positivity-preserving DIRK-LDG discretizations for problems requiring a positivity limiter. In addition, we can observe from the numerical results that the implicit time-discrete methods alleviate the time-step restrictions needed for the stability of the numerical discretizations, which improves computational efficiency.

An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension

An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension

A sixth-order finite difference HWENO scheme for nonlinear degenerate parabolic equation

A sixth-order finite difference HWENO scheme for nonlinear degenerate parabolic equation

EnVarA-FEM for the flux-limited porous medium equation

The flux-limited porous medium equation is a kind of nonlinear degenerate parabolic equation based on the porous medium equation, its solution not only has a finite propagation speed and possible waiting time, but also may develop jump fronts. It is difficult to construct numerical methods to capture these features in an effective way. We propose structure-preserving finite element numerical schemes based on the energetic variational approach, which are proved to be uniquely solvable, positivity preserving and energy dissipative. The analysis of the free boundaries conditions is given to illustrate the circumstances under which the solution of the flux-limited porous medium equation will have finite propagation speed, waiting time or jump fronts. Moreover, the optimal convergence analysis for the nonlinear numerical scheme is presented by applying the technique of combining the rough and refined error estimates. Numerical experiments reveal that the proposed numerical method can successfully capture important features of the flux-limited PME: finite propagation speed of the free boundary, waiting time, effect of amplitudes on the regularity of the solutions and jump fronts. In particular, the waiting time is computed naturally and the convergence order of the waiting time can be given numerically. A criterion to approximate the time where discontinuous interfaces occur is also established, and the corresponding convergence order is given numerically.

On the theory of entropy suband supersolutions of nonlinear degenerate parabolic equations

We consider a second-order nonlinear degenerate anisotropic parabolic equation in the case when the flux vector is only continuous and the nonnegative diffusion matrix is bounded and measurable. The concepts of entropy sub- and supersolution of the Cauchy problem are introduced, so that the entropy solution of this problem, understood in the sense of Chen-Perthame, is both an entropy sub- and supersolution. It is established that the maximum of entropy subsolutions of the Cauchy problem is also an entropy subsolution of this problem. This result is used to prove the existence of the largest entropy subsolution (and the smallest entropy supersolution). It is also shown that the largest entropy subsolution and the smallest entropy supersolution are also entropy solutions.

A sixth-order central WENO scheme for nonlinear degenerate parabolic equations

In this paper, we develop a new sixth-order finite difference central weighted essentially non-oscillatory (WENO) scheme with Z-type nonlinear weights for nonlinear degenerate parabolic equations. The centered polynomial is introduced for the WENO reconstruction to avoid the negative linear weights. We choose the Z-type nonlinear weights based on the $$L^2$$ -norm smoothness indicators, yielding the new WENO scheme with less computational cost. It is also confirmed that the proposed central WENO scheme with the devised nonlinear weights achieves sixth order accuracy in smooth regions. One- and two-dimensional numerical examples are presented to demonstrate the improved performance of the proposed central WENO scheme.

A sixth-order finite difference WENO scheme for non-linear degenerate parabolic equations: an alternative technique

PurposeThis paper aims to construct a sixth-order weighted essentially nonoscillatory scheme for simulating the nonlinear degenerate parabolic equations in a finite difference framework.Design/methodology/approachTo design this scheme, we approximate the second derivative in these equations in a different way, which of course is still in a conservative form. In this way, unlike the common practice of reconstruction, the approximation of the derivatives of odd order is needed to develop the numerical flux.FindingsThe results obtained by the new scheme produce less error compared to the results of other schemes in the literature that are recently developed for the nonlinear degenerate parabolic equations while requiring less computational times.Originality/valueThis research develops a new weighted essentially nonoscillatory scheme for solving the nonlinear degenerate parabolic equations in multidimensional space. Besides, any selection of the constants (sum equals one is the only requirement for them), named the linear weights, will obtain the desired accuracy.

Nonlinear Einstein paradigm of Brownian motion and localization property of solutions

We employ a generalization of Einstein's random walk paradigm for diffusion to derive a class of multidimensional degenerate nonlinear parabolic equations in nondivergence form. Specifically, in these equations, the diffusion coefficient can depend on both the dependent variable and its gradient, and it vanishes when either one of the latter does. It is known that solutions of such degenerate equations can exhibit finite speed of propagation (so‐called localization property of solutions). We give a proof of this property using a De Giorgi–Ladyzhenskaya iteration procedure for nondivergence form equations. A mapping theorem is then established to a divergence‐form version of the governing equation for the case of one spatial dimension. Numerical results via a finite‐difference scheme are used to illustrate the main mathematical results for this special case. For completeness, we also provide an explicit construction of the one‐dimensional self‐similar solution with finite speed of propagation function, in the sense of Kompaneets–Zel'dovich–Barenblatt. We thus show how the finite speed of propagation quantitatively depends on the model's parameters.

An oscillation free local discontinuous Galerkin method for nonlinear degenerate parabolic equations

AbstractAn oscillation free local discontinuous Galerkin (OFLDG) method is proposed for solving nonlinear degenerate parabolic equations. The damping terms are added to the original LDG scheme to control the spurious oscillations when solutions have a large gradient. The L2‐stability and optimal priori error estimates for the semi‐discrete scheme in one‐ and multi‐dimensions are established. The numerical experiments demonstrate that the proposed method maintains the high order accuracy and controls the spurious oscillations well.

Entropy Formulation for Triply Nonlinear Degenerate Elliptic-Parabolic-Hyperbolic Equation with Zero-Flux Boundary Condition

In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.

Optimal subalgebras and conservation laws with exact solutions for biological population model

In the present study, we focus on the (2+1) dimensional normal biological population model (NBPM), which describes the population migration of species. We employ Lie symmetry analysis to the given nonlinear degenerate parabolic partial differential equation (PDE), which shows substantial advancement and upgraded results over other analytical techniques in determining some classes of exact solutions. Using the symmetry group of transformations, we construct the one-dimensional and two-dimensional optimal subalgebras for the NBPM. Further, we present the reduced ordinary differential equation(ODE) for each one-dimensional optimal subalgebras and construct some exact solutions for the physical model. Furthermore, we illustrate the physical behaviour of the model graphically through the obtained exact solutions. Lastly, applying the multipliers method, we develop some new conservation laws yielding some potential systems which are nonlocally related to the given PDE.

Velocity averaging for diffusive transport equations with discontinuous flux

We consider a diffusive transport equation with discontinuous flux and prove the velocity averaging result under non-degeneracy conditions. In order to achieve the result, we introduce a new variant of micro-local defect functionals which are able to ‘recognise’ changes of the type of the equation. As a corollary, we show the existence of a weak solution for the Cauchy problem for non-linear degenerate parabolic equation with discontinuous flux. We also show existence of strong traces at t = 0 $t=0$ for so-called quasi-solutions to degenerate parabolic equations under non-degeneracy conditions on the diffusion term.

A RBF-WENO Finite Difference Scheme for Non-linear Degenerate Parabolic Equations

A RBF-WENO Finite Difference Scheme for Non-linear Degenerate Parabolic Equations

A new type of high-order finite difference compact reconstruction multi-resolution WENO scheme for nonlinear degenerate parabolic equations

A new type of high-order finite difference compact reconstruction multi-resolution WENO scheme for nonlinear degenerate parabolic equations

On decay of entropy solutions to degenerate nonlinear parabolic equations with perturbed periodic initial data

Under a precise nonlinearity-diffusivity assumption we establish the decay of entropy solutions of a degenerate nonlinear parabolic equation with initial data being a sum of periodic function and a function vanishing at infinity in the appropriate sense.

A New Sixth-Order Finite Difference Compact Reconstruction Unequal-Sized WENO Scheme for Nonlinear Degenerate Parabolic Equations

In this paper, a new sixth-order finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory (CRUS-WENO) scheme is designed for solving the nonlinear degenerate parabolic equations on structured meshes. This new CRUS-WENO scheme only uses the information defined on three unequal-sized spatial stencils, obtains the optimal sixth-order accuracy in smooth regions, and preserves the second-order accuracy near strong discontinuities. This scheme can be applied to dispose the emergence of the negative linear weights and avoid the application of the mapped function. The corresponding linear weights can be artificially set to be any random positive numbers as well as their summation is one. The construction process of this scheme is very simple and can be easily extended to higher dimensions, since the new compact conservative formulation is applied to approximate the second-order derivatives. The new CRUS-WENO scheme uses narrower large stencil than that of the same order finite difference classical WENO schemes. Therefore, it has a better compactness and smaller truncation errors. Some benchmark numerical tests including the porous medium equation and the degenerate parabolic convection-diffusion equation are performed to illustrate the advantages of this new CRUS-WENO scheme.