Derivation of the discrete conservation laws for a family of finite difference schemes

1994 ◽  
Vol 64 (1) ◽  
pp. 13-45 ◽  
Author(s):  
Salvador Jiménez
Keyword(s):  
Finite Difference ◽  
Conservation Laws ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Discrete Conservation Laws

COMPARISON OF FINITE‐DIFFERENCE SCHEMES FOR THE GROSS‐PITAEVSKII EQUATION

2009 ◽  
Vol 14 (1) ◽  
pp. 109-126 ◽  
Author(s):  
Vyacheslav A. Trofimov ◽  
Nikolai Peskov
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Conservation Laws ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Pitaevskii Equation ◽  
Conservative Scheme ◽  
Conservative Finite Difference Scheme

A conservative finite‐difference scheme for numerical solution of the Gross‐Pitaevskii equation is proposed. The scheme preserves three invariants of the problem: the L 2 norm of the solution, the impulse functional, and the energy functional. The advantages of the scheme are demonstrated via several numerical examples in comparison with some other well‐known and widely used methods. The paper is organized as follows. In Section 2 we consider three main conservation laws of GPE and derive the evolution equations for first and second moments of a solution of GPE. In Section 3 we define the conservative finite‐difference scheme and prove the discrete analogs of conservation laws. The remainder of Section 3 consists of a brief description of other finite‐difference schemes, which will be compared with the conservative scheme. Section 4 presents the results of numerical solutions of three typical problems related to GPE, obtained by different methods. Comparison of the results confirms the advantages of conservative scheme. And finally we summarize our conclusions in Section 5.

scholarly journals Simple bespoke preservation of two conservation laws

2018 ◽  
Vol 40 (2) ◽  
pp. 1294-1329 ◽  
Author(s):  
Gianluca Frasca-Caccia ◽  
Peter Ellsworth Hydon
Keyword(s):  
Finite Difference ◽  
Conservation Laws ◽  
Vries Equation ◽  
Finite Difference Methods ◽  
Nonlinear Heat Equation ◽  
Finite Difference Schemes ◽  
Simplified Approach ◽  
Numerical Tests ◽  
Difference Methods ◽  
Discrete Conservation Laws

Abstract Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of the Euler operator, an observation that was first used recently to construct approximations symbolically that preserve two conservation laws of a given PDE. However, the complexity of the symbolic computations has limited the effectiveness of this approach. The current paper introduces some key simplifications that make the symbolic–numeric approach feasible. To illustrate the simplified approach we derive bespoke finite difference schemes that preserve two discrete conservation laws for the Korteweg–de Vries equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared with others in the literature.

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