scholarly journals Convergence of fully discrete schemes for diffusive dispersive conservation laws with discontinuous coefficient

2016 ◽  
Vol 50 (5) ◽  
pp. 1289-1331
Author(s):  
Rajib Dutta ◽  
Ujjwal Koley ◽  
Deep Ray
Keyword(s):  
Conservation Laws ◽  
Discontinuous Coefficient ◽  
Fully Discrete

Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems

Acta Numerica ◽  
2003 ◽  
Vol 12 ◽  
pp. 451-512 ◽  
Author(s):  
Eitan Tadmor
Keyword(s):  
Conservation Laws ◽  
Entropy Production ◽  
Time Dependent ◽  
Closed Form Expression ◽  
Entropy Stability ◽  
Fully Discrete ◽  
Spatial Entropy ◽  
Difference Approximations ◽  
Conservative Schemes ◽  
Time Dependent Problems

We study the entropy stability of difference approximations to nonlinear hyperbolic conservation laws, and related time-dependent problems governed by additional dissipative and dispersive forcing terms. We employ a comparison principle as the main tool for entropy stability analysis, comparing the entropy production of a given scheme against properly chosen entropy-conservative schemes.To this end, we introduce general families of entropy-conservative schemes, interesting in their own right. The present treatment of such schemes extends our earlier recipe for construction of entropy-conservative schemes, introduced in Tadmor (1987b). The new families of entropy-conservative schemes offer two main advantages, namely, (i) their numerical fluxes admit an explicit, closed-form expression, and (ii) by a proper choice of their path of integration in phase space, we can distinguish between different families of waves within the same computational cell; in particular, entropy stability can be enforced on rarefactions while keeping the sharp resolution of shock discontinuities.A comparison with the numerical viscosities associated with entropy-conservative schemes provides a useful framework for the construction and analysis of entropy-stable schemes. We employ this framework for a detailed study of entropy stability for a host of first- and second-order accurate schemes. The comparison approach yields a precise characterization of the entropy stability of semi-discrete schemes for both scalar problems and systems of equations.We extend these results to fully discrete schemes. Here, spatial entropy dissipation is balanced by the entropy production due to time discretization with a suffciently small time-step, satisfying a suitable CFL condition. Finally, we revisit the question of entropy stability for fully discrete schemes using a different approach based on homotopy arguments. We prove entropy stability under optimal CFL conditions.

scholarly journals A relaxation scheme for conservation laws with a discontinuous coefficient

2003 ◽  
Vol 73 (247) ◽  
pp. 1235-1260 ◽  
Author(s):  
K. H. Karlsen ◽  
C. Klingenberg ◽  
N. H. Risebro
Keyword(s):  
Conservation Laws ◽  
Discontinuous Coefficient ◽  
Relaxation Scheme

Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws

2019 ◽  
Vol 53 (1) ◽  
pp. 105-144 ◽  
Author(s):  
Lingling Zhou ◽  
Yinhua Xia ◽  
Chi-Wang Shu
Keyword(s):  
Conservation Laws ◽  
Galerkin Method ◽  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Second Order ◽  
Arbitrary Lagrangian Eulerian ◽  
Third Order ◽  
Fully Discrete ◽  
Runge Kutta

In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh2 and the second order TVD-RK scheme needs $ \tau \le \rho {h}^{\frac{4}{3}}$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h.

DISCRETE COMPRESSIVE SOLUTIONS OF SCALAR CONSERVATION LAWS

2004 ◽  
Vol 01 (03) ◽  
pp. 493-520
Author(s):  
BRUNO DESPRÉS
Keyword(s):  
Theoretical Analysis ◽  
Conservation Laws ◽  
Rate Of Convergence ◽  
Numerical Experiments ◽  
Scalar Conservation Laws ◽  
Initial Profile ◽  
Fully Discrete ◽  
Discrete Approach ◽  
Scalar Conservation ◽  
Proof Of Convergence

We prove the convergence of numerical approximations of compressive solutions for scalar conservation laws with convex flux. This new proof of convergence is fully discrete and does not use Kuznetsov's approach. We recover the well-known rate of convergence in O(Δx½). With the same fully discrete approach, we also prove a rate of convergence in O(Δx) uniformly in time, if the initial data is a shock, or asymptotically after the compression of the initial profile. Numerical experiments confirm the theoretical analysis.

ENTROPY CONSERVATIVE SCHEMES AND ADAPTIVE MESH SELECTION FOR HYPERBOLIC CONSERVATION LAWS

2010 ◽  
Vol 07 (03) ◽  
pp. 383-404 ◽  
Author(s):  
CHRISTOS ARVANITIS ◽  
CHARALAMBOS MAKRIDAKIS ◽  
NIKOLAOS I. SFAKIANAKIS
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Adaptive Mesh ◽  
Mesh Adaptivity ◽  
Hyperbolic Conservation ◽  
Fully Discrete ◽  
Conservative Schemes ◽  
Mesh Selection ◽  
Adaptive Schemes ◽  
Selection For

We consider numerical schemes which combine non-uniform, adaptively redefined spatial meshes with entropy conservative schemes for the evolution step for shock computations. We observe that the resulting adaptive schemes yield approximations free of oscillations in contrast to known fully discrete entropy conservative schemes on uniform meshes. We conclude that entropy conservative schemes are transformed to entropy diminishing schemes when combined with the proposed geometrically driven mesh adaptivity.

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