On the convergence of operator splitting applied to conservation laws with source terms

@article{Langseth1996OnTC,
  title={On the convergence of operator splitting applied to conservation laws with source terms},
  author={Jan Olav Langseth and Aslak Tveito and Ragnar Winther},
  journal={SIAM Journal on Numerical Analysis},
  year={1996},
  volume={33},
  pages={843-863}
}
The order of convergence for operator splitting applied to conservation laws with source terms is studied. The operator splitting procedure is based on local solutions of the associated homogeneous conservation law and an ordinary differential equation. We prove that, for scalar problems, the error introduced by the splitting is linear with respect to the time step. The theoretical results are illustrated by numerical examples. 

Figures from this paper

Splitting method applied to hyperbolic problem with source term

  • F. Peyroutet
  • Computer Science, Mathematics
    Appl. Math. Lett.
  • 2001

ERROR ESTIMATE FOR A SPLITTING METHOD APPLIED TO CONVECTION-REACTION EQUATIONS

An operator splitting method is used to approximate solutions of initial–boundary value problems related to a hyperbolic reaction equation. In this paper, we study the order of convergence for this

A Time-Fractional Step Method for Conservation Law Related Obstacle Problems

Here, one proves that this time-splitting method developed classically to compute discontinuous solutions of nonhomogeneous scalar conservation laws converges in L^1 to the weak entropy solution of the considered obstacle problem.

Convergence Analysis for Operator Splitting Methods toConservation Laws with Sti Source

  • T. Tang
  • Mathematics, Computer Science
  • 1996
It is proved that the L 1 error introduced by the time-splitting can be bounded by O((tkq(u 0)k L 1 (R), which is an improvement of the O(Qt) upper bound, where t is the splitting time step.

Convergence Analysis for Operator Splitting Methods to Conservation Laws with Stii Source Terms

We analyze the order of convergence for operator splitting methods applied to conservation laws with stii source terms. We suppose that the source term q(u) is dissipative. It is proved that the L 1

Equilibrium schemes for scalar conservation laws with stiff sources

A kinetic interpretation of upwinding taking into account the source terms of scalar conservation laws with a zeroth order source with low regularity and a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions.

Error estimators for the position of discontinuities in hyperbolic conservation laws with source terms which are solved using operator splitting

This paper derives error-estimates for the scalar Riemann problem and applies them to an adaptation of the step size so that the error of the location of the discontinuity remains sufficiently small.

Convergence Analysis for Operator-Splitting Methods Applied to Conservation Laws with Stiff Source Terms

We analyze the order of convergence for operator splitting methods applied to conservation laws with stiff source terms. We suppose that the source term $q(u)$ is dissipative. It is proved that the

Operator-Splitting on Hyperbolic Balance Laws

This work will begin with the analysis of the splitting of multi-dimensional linear systems and end up explaining how exact nonlinear splitting can be obtained for one-dimensional scalar equations.

References

SHOWING 1-10 OF 24 REFERENCES

The method of fractional steps for conservation laws

It is proved that both first order splitting and Strang splitting algorithms always converge to the unique weak solution satisfying the entropy condition.

A moving mesh numerical method for hyperbolic conservation laws

We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in L1(R) to within O(N-2) by a piecewise linear function with O(fN) nodes; the

A Conservative Front Tracking Scheme for 1D Hyperbolic Conservation Laws

A front tracking scheme for one dimensional hyperbolic systems of conservation laws is reviewed. The scheme is applied to the equations of gas-dynamics and a non-strictly hyperbolic system of

Front Tracking Applied to a Nonstrictly Hyperbolic System of Conservation Laws

The application of a front tracking method to a nonstrictly hyperbolic system of conservation laws is described in one space dimension and the method is compared with the random choice scheme and the upwind scheme.

A method of fractional steps for scalar conservation laws without the CFL condition

We present a numerical method for the n-dimensional initial value problem for the scalar conservation law u(xl. x, , t)I + Z7 f1(u)x = 0, u(x . x, , 0) = uo(x . xv) . Our method is based on the use

Conservation laws with a random source

We study the scalar conservation law with a noisy nonlinear source, namely,ul + f(u)x = h(u, x, t) + g(u)W(t), whereW(t) is the white noise in the time variable, and we analyse the Cauchy problem for

A front-tracking alternative to the random choice method

An alternative to Glimm's proof of the existence of solutions to systems of hyperbolic conservation laws is presented. The proof is based on an idea by Dafermos for the single conservation law and in