scholarly journals Hyperbolic Conservation Laws on Manifolds: Total Variation Estimates and the Finite Volume Method

2005 ◽  
Vol 12 (3) ◽  
pp. 291-324 ◽  
Author(s):  
Paulo Amorim ◽  
Matania Ben-Artzi ◽  
Philippe G. LeFloch
Keyword(s):  
Conservation Laws ◽  
Finite Volume Method ◽  
Total Variation ◽  
Finite Volume ◽  
Hyperbolic Conservation Laws ◽  
Hyperbolic Conservation ◽  
Volume Method

scholarly journals Finite volume method in curvilinear coordinates for hyperbolic conservation laws

2011 ◽  
Vol 32 ◽  
pp. 163-176 ◽  
Author(s):  
A. Bonnement ◽  
T. Fajraoui ◽  
H. Guillard ◽  
M. Martin ◽  
A. Mouton ◽  
...  
Keyword(s):  
Conservation Laws ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Hyperbolic Conservation Laws ◽  
Curvilinear Coordinates ◽  
Hyperbolic Conservation ◽  
Volume Method

scholarly journals A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter

2015 ◽  
Vol 18 (4) ◽  
pp. 901-930 ◽  
Author(s):  
Ziyao Sun ◽  
Honghui Teng ◽  
Feng Xiao
Keyword(s):  
Conservation Laws ◽  
Finite Volume ◽  
Solution Structure ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Weno Reconstruction ◽  
Hyperbolic Conservation ◽  
Average Value ◽  
High Order Schemes ◽  
Volume Method

AbstractThis paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.

scholarly journals Convergence of the finite volume method on a Schwarzschild background

2019 ◽  
Vol 53 (5) ◽  
pp. 1459-1476
Author(s):  
Shijie Dong ◽  
Philippe G. LeFloch
Keyword(s):  
Black Hole ◽  
Conservation Laws ◽  
Finite Volume ◽  
Hyperbolic Conservation Laws ◽  
Hyperbolic Conservation ◽  
New Class ◽  
Black Hole Background ◽  
Existence And Stability ◽  
Volume Method ◽  
Nonlinear Hyperbolic Conservation Laws

We introduce a class of nonlinear hyperbolic conservation laws on a Schwarzschild black hole background and derive several properties satisfied by (possibly weak) solutions. Next, we formulate a numerical approximation scheme which is based on the finite volume methodology and takes the curved geometry into account. An interesting feature of our model is that no boundary conditions is required at the black hole horizon boundary. We establish that this scheme converges to an entropy weak solution to the initial value problem and, in turn, our analysis also provides us with a theory of existence and stability for a new class of conservation laws.

scholarly journals A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws

2018 ◽  
Vol 40 (1) ◽  
pp. 405-421 ◽  
Author(s):  
N Chatterjee ◽  
U S Fjordholm
Keyword(s):  
Initial Data ◽  
Conservation Laws ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Numerical Examples ◽  
Continuity Equations ◽  
Multiple Dimensions ◽  
Nonlocal Conservation Laws ◽  
Volume Method ◽  
Singular Data

Abstract We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

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