An investigation of postshock oscillations on non-uniform grids is performed in this paper. These oscillations are generated as shock passes through the grid interfaces. The LLF scheme is checked for 1 D and 2 D problems on the discontinuous grids. Oscillations are observed only for nonlinear systems and the solutions of the scalar conservation laws and linear systems behave logically. The integral curves suggest underlying properties of these oscillations. The results of the paper reveal a flaw that adaptive methods for conservation laws have to refine grids at each time step.
This paper considers the numerical solutions of hyperbolic systems of conservation laws on discontinuous grids. The discontinuity in the grid is often due to the overlapping of different mesh systems, or to the adaptive mesh refinement. In either case, if the discontinuities in solutions are genuine nonlinear, some oscillations will be conserved around the grid interface for nonlinear systems.
Computations on discontinue grids are becoming more common for two reasons. First, many multidimensional problems of practice interest involve complex geometry, and in general it is not sufficient to be able solve hyperbolic equations on a uniform Cartesian grid in a rectangular domain. As the configurations that can be modeled become more complex, so does the grid generation problem. Generally, it is very difficult to generate one smooth body-fitted grid to cover the whole complex domain. To simplify this procedure, it is accustomed to use multigrid to fit the complex domain. Each part of the domain will still have a smooth grid, but now the component grids will in general overlap rather than in an irregular fashion.
The second reason of these discontinue grids comes from the use of adaptive methods [
The main objective of this paper is to investigate postshock oscillations around the grid interface. Some linear and nonlinear equations are tested with LLF scheme, and this kind of oscillations only appears in nonlinear equation system. In addition, Godunov scheme behaves similarly to the LLF scheme, and so we don’t give the numerical results of Godunov scheme for saving space. We incline to believe that the postshock oscillations are an inevitable feature of shocks captured by currently frequently-used methods. The postshock oscillations due to discontinue grids are different from common nonphysical oscillations caused by high-order interpolation, but it is very similar to that appear behind the slow shocks [
The rest of the paper is organized as follows. In Section 2, we briefly review some model problems and give the mesh distribution. In Section 3, the LLF scheme is described. The numerical results and the observed behavior are demonstrated in Section 4. However, we have yet found the reason which leads to the postshock oscillations observed in the paper.
The postshock oscillations are investigated under the following five sets of equations: the advection equation,
u t + u x = 0 ; (1)
the inviscid Burgers equation,
u t + ( u 2 2 ) x = 0 ; (2)
the shallow water equation,
[ h h u ] t + [ h u h u 2 + 1 2 g h 2 ] x = 0 , (3)
where h is the depth of the fluid, u is the horizontal velocity and g is acceleration of gravity; and full Euler equations for an ideal gas with constant specific heats,
[ ρ ρ u E ] t + [ ρ u ρ u 2 + p u ( E + p ) ] x = 0 , (4)
where ρ , u , E , p are the density, velocity, total energy and pressure, respectively. The system of Euler equations is closed by the equation of state for an ideal polytropic gas
E = p γ − 1 + 1 2 ρ u 2 ,
here γ = 1.4 is the ratio of specific heat.
The last example is 2D Euler equations,
[ ρ ρ u ρ v E ] t + [ ρ u ρ u 2 + p ρ u v u ( E + p ) ] x + [ ρ v ρ u v ρ v 2 + p v ( E + p ) ] y = 0 , (5)
where the above system is closed by the equation of state,
E = p γ − 1 + 1 2 ρ ( u 2 + v 2 ) .
For ease of presentation, we consider the case of mesh refinement by a factor θ .
Δ x 1 = ⋯ = Δ x N = b − a N + M θ and Δ x N + 1 = ⋯ = Δ x N + M = b − a N + M θ θ . Apparently, the interface between the coarse and fine grids is located at x N + 1 2 .
The equations presented in the previous section will be solved using first order accurate LLF [
U i n + 1 = U i n − Δ t Δ x ( F i + 1 / 2 n − F i − 1 / 2 n ) , (6)
where F j + 1 / 2 n is the numerical flux function at the interface between cells i and i + 1 at time level n . Depending on the choice of the numerical flux formula, Equation (6) is referred to as LLF scheme.
LLF scheme is the improved version of the classical Lax-Friedrichs scheme by replacing the value a = Δ t Δ x by a locally determined value,
F i + 1 / 2 = 1 2 [ F ( U i ) + F ( U i + 1 ) − a i + 1 / 2 ( U i + 1 − U i ) ] (7)
where a i + 1 / 2 = max ( | λ ( U i ) | , | λ ( U i + 1 ) | ) .
Firstly, advection Equation (1) is considered on a non-uniform grid which is composed of two uniform subgrids with initial condition
u ( x , 0 ) = { 1, if − 2 ≤ x ≤ − 1, 0, if − 1 ≤ x ≤ 1.
The ratio between the coarse and fine grids is θ = 2 , and the interface in the physics domain is located at x = 0 . The boundary of domain a = − 2 , b = 1 , and the number of cell N = 200 , M = 200 . Obviously, the discontinuity will arrive the interface when t = 1 and wholly pass through it when t = 1.5 .
For the linear advection equation, LLF scheme reduces to upwind scheme. The solutions to advection equation can be seen in
The second example is the inviscid Burgers Equation (2) subject to the initial data
u ( x , 0 ) = { 2, if − 2 ≤ x ≤ − 1, 0, if − 1 ≤ x ≤ 1.
The solution propagates to the right with shock speed s = 1 .
Consider the shallow water equations (3) with the piecewise-constant initial conditions
h ( x , 0 ) = { 3 , if − 5 ≤ x ≤ 0 , 1 , if 0 ≤ x ≤ 5 , u ( x , 0 ) = 0. (8)
This is a special case of the Riemann problem in which u l = u r = 0 , and is called the dam-break problem because it models what happens if a dam separating two levels of water bursts at time t = 0 . This is the shallow water equivalent of the shock tube problem of gas dynamics.
Note that the Jacobian matrix F ′ ( u ) of the shallow water equations is
F ′ ( u ) = [ 0 1 − u 2 + g h 2 u ] .
The eigenvalues of F ′ ( u ) are
λ 1 = u − g h , λ 2 = u + g h ,
with the corresponding eigenvectors
r 1 = [ 1 u − g h ] , r 2 = [ 1 u + g h ] .
Firstly, a uniform grid is considered.
For the shallow water equations, Leveque [
loci and integral curves. The Hugoniot curves correspond to the 1-shock and 2-shock are
h u = u l − h g 2 ( h l h − h h l ) ( h l − h )
and
h u = u r + h g 2 ( h r h − h h r ) ( h r − h )
respectively.
The integral curves correspond to 1-rarefaction and 2-rarefaction waves are
h u = h u l + 2 h ( g h l − g h )
and
h u = h u r − 2 h ( g h r − g h )
respectively.
As shown in previous paragraph, the solution of this initial-value problem contains both shock and rarefaction. So, Hugoniot and integral curves originate from the left state U r and right state U l , and the intersection of them is intermediate state U m .
In
after it pass through the interface. The oscillation is the wave that arose from the initial discontinuity at x = 3 when the shocks pass the interface.
A interesting phenomenon can be observed in
Note that, for the shallow water equations, the function of 1-Riemann invariant and 2-Riemann invariant are
w 1 = u + 2 g h
and
w 2 = u − 2 g h
respectively.
Euler equations of gas dynamics is discussed in this section. We test the postshock oscillations with initial data
( ρ , v , p ) = { ( 1 , 0 , 1 ) , if x ≤ 0 , ( 0.125 , 0 , 0.1 ) , if x > 0. (9)
This is a well known test problem proposed by Sod [
[ − 5,5 ] is also divided into two uniform subdomain [ − 5,3 ] and [ 3,5 ] .
Except for the first and third characteristic fields are genuinely nonlinear which have similar behavior to the two characteristic fields in the shallow water equations, the second field is linearly degeneration. So, the solution to Riemann problem for Euler problem typically has two nonlinear waves and a contact discontinuity.
From
For this initial value problem, all of the Riemann invariants for a polytropic ideal gas are summarized below:
1 -Riemann invariants , s , u + 2 c γ − 1 , 2 -Riemann invariants , u , p , 3 -Riemann invariant , u − 2 c γ − 1 .
find that the 1-Riemann invariants are constant across the rarefaction and the left wave which generate from x = 3 , and only one oscillation was left. Similarly,
The last example is 2D Euler equations (5), with initial data
( ρ , u , v , p ) = { ( 1.1 , 0.0 , 0.0 , 1.1 ) , if x > 0.5 , y > 0.5 , ( 0.5065 , 08939 , 0.0 , 0.35 ) , if x < 0.5 , y > 0.5 , ( 1.1 , 0.8939 , 0.8939 , 1.1 ) , if x < 0.5 , y < 0.5 , ( 0.5065 , 0.0 , 0.8939 , 0.35 ) , if x > 0.5 , y < 0.5 , (10)
which corresponds to the case of left forward shock, right backward shock, upper backward shock, and lower forward shock. We refer the readers to [
The postshock oscillations appeared in the nonlinear system has been elucidated numerically. We found that these oscillations would not appear in the nonlinear scalar equation and linear equations system. Although all the numerical results were computed with LLF method, these oscillations also appeared with other schemes, such as Godunov, Roe schemes and so on. So the postshock oscillations
have nothing to do with the specific methods which are used to compute solution on nonuniform grids. In addition, this kind of oscillations is different from those nonphysical oscillations arose near the shock, and they are similar to those emerge behind the slow-moving shocks which can be found in [
This work was supported by the National Natural Science Foundation of China (No. 11501238), Natural Science Foundation of Guangdong Province (No. 2012A030313119) and Supported by the Major Project Foundation of Guangdong Province Education Department (No. 2014KZDXM070).
Hu, F.X. (2017) Postshock Oscillations on Non-Uniform Mesh. Journal of Applied Mathematics and Physics, 5, 481-493. https://doi.org/10.4236/jamp.2017.52043