scholarly journals A New High-Order Approximation for the Solution of Two-Space-Dimensional Quasilinear Hyperbolic Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
R. K. Mohanty ◽  
Suruchi Singh
Keyword(s):  
Dirichlet Boundary ◽  
Hyperbolic Equations ◽  
Order Approximation ◽  
High Order ◽  
Dirichlet Boundary Conditions ◽  
Polar Coordinates ◽  
Hyperbolic Partial Differential Equation ◽  
High Order Approximation ◽  
Linear Hyperbolic Equation ◽  
Derivation Procedure

we propose a new high-order approximation for the solution of two-space-dimensional quasilinear hyperbolic partial differential equation of the form , , , subject to appropriate initial and Dirichlet boundary conditions , where and are mesh sizes in time and space directions, respectively. We use only five evaluations of the function as compared to seven evaluations of the same function discussed by (Mohanty et al., 1996 and 2001). We describe the derivation procedure in details and also discuss how our formulation is able to handle the wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Some examples and their numerical results are provided to justify the usefulness of the proposed method.

High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Jianxiong Cao ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Dirichlet Boundary ◽  
Order Approximation ◽  
High Order ◽  
Dirichlet Boundary Conditions ◽  
High Order Accuracy ◽  
High Order Approximation ◽  
Order Accuracy ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
New Formula

AbstractIn this paper, we first establish a high-order numerical algorithm for α-th (0 < α < 1) order Caputo derivative of a given function f(t), where the convergence rate is (4 − α)-th order. Then by using this new formula, an improved difference scheme with high order accuracy in time to solve Caputo-type fractional advection-diffusion equation with Dirichlet boundary conditions is constructed. Finally, numerical examples are carried out to confirm the efficiency of the constructed algorithm.

HIGH ACCURACY ARITHMETIC AVERAGE TYPE DISCRETIZATION FOR THE SOLUTION OF TWO-SPACE DIMENSIONAL NONLINEAR WAVE EQUATIONS

2012 ◽  
Vol 03 (02) ◽  
pp. 1150005 ◽  
Author(s):  
R. K. MOHANTY ◽  
VENU GOPAL
Keyword(s):  
Differential Equation ◽  
Wave Equations ◽  
Dirichlet Boundary ◽  
High Accuracy ◽  
Dirichlet Boundary Conditions ◽  
Polar Coordinates ◽  
Nonlinear Wave Equations ◽  
Hyperbolic Partial Differential Equation ◽  
Unconditionally Stable ◽  
Arithmetic Average

In this paper, we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyperbolic partial differential equation of the form utt = A(x, y, t)uxx + B(x, y, t)uyy + g(x, y, t, u, ux, uy, ut), 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions. We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation. The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable. Numerical results are provided to illustrate the usefulness of the proposed method.

The SKN Approximation for Solving Radiative Transfer Problems in Absorbing, Emitting, and Isotropically Scattering Plane-Parallel Medium: Part 1

2002 ◽  
Vol 124 (4) ◽  
pp. 674-684 ◽  
Author(s):  
Zekeriya Altac¸
Keyword(s):  
Integral Equation ◽  
Radiative Transfer ◽  
Differential Equations ◽  
Order Approximation ◽  
High Order ◽  
High Order Approximation ◽  
One Dimensional ◽  
Second Order Differential Equations ◽  
Plane Parallel ◽  
Transfer Problems

A high order approximation, the SKN method—a mnemonic for synthetic kernel—is proposed for solving radiative transfer problems in participating medium. The method relies on approximating the integral transfer kernel by a sum of exponential kernels. The radiative integral equation is then reducible to a set of coupled second-order differential equations. The method is tested for one-dimensional plane-parallel participating medium. Three quadrature sets are proposed for the method, and the convergence of the method with the proposed sets is explored. The SKN solutions are compared with the exact, PN, and SN solutions. The SK1 and SK2 approximations using quadrature Set-2 possess the capability of solving radiative transfer problems in optically thin systems.

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