Stiff ode15s Simulation of a Simple Three-Species Oscillating Chemical Reaction Model

Project info

• Field of studyChemistry and Materials
• LevelDoctoral dissertation (S3)
• Method typeNumerical Simulation
• Estimated build time± 7 working days
• Toolsode15s, ode45, odeset (MATLAB inti)
• PriceUS$41 - US$253

Research goal

Simulate intermediate species concentration in a stiff system with ode15s and assess the influence of fast rate parameter and slow rate parameter.

Variables

Example result

• 58Stiffness parameter mu
• 5645ode45 steps
• 678ode15s steps
• 8.326Step ratio
• 36433ode45 f-evals
• 1716ode15s f-evals

For the stiff Intermediate species concentration system (mu=58) ode15s needs only 678 steps versus 5645 for ode45 (a 8.3x ratio), with a mean solution difference of 0.002.

Computed from illustrative synthetic data, not from a real study.

About Stiff Systems with ode15s

ode15s is a solver for stiff differential equation systems, whose widely different time scales force ordinary solvers to take very small steps. It is used in chemical kinetics, electronic circuits, and process models with fast and slow dynamics.

Steps

1. 1Recognise stiffness: ode45 is very slow or takes tiny steps
2. 2Write the system as dy/dt = f(t,y)
3. 3Solve with ode15s and set tolerances
4. 4Also solve with ode45 as a comparison
5. 5Compare the number of steps and computation time
6. 6Plot the solutions and step counts

Assumptions to check

• The system is really stiff (large time-scale ratio)
• The Jacobian can be estimated or supplied for efficiency
• Tolerances are chosen according to accuracy needs

Reading and reporting the results

Report the number of steps and run time of ode15s versus ode45 as evidence of stiffness. Explain the fast and slow time scales in the model.

Example MATLAB code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

mu = 200;                                          % Van der Pol stiffness parameter
f = @(t, y) [y(2); mu*(1 - y(1)^2)*y(2) - y(1)];
tspan = [0 400];
y0 = [2; 0];
opts = odeset('RelTol', 1e-6, 'AbsTol', 1e-8);

tic; [t15, y15] = ode15s(f, tspan, y0, opts); t15s = toc;
tic; [t45, y45] = ode45(f, tspan, y0, opts); t45s = toc;
fprintf('ode15s: %d steps, %.3f s\n', numel(t15) - 1, t15s);
fprintf('ode45 : %d steps, %.3f s\n', numel(t45) - 1, t45s);

fast_rate_parameter = t15;                                      % time vector
intermediate_species_concent = y15;                                       % states
figure;
subplot(2,1,1);
plot(fast_rate_parameter, intermediate_species_concent(:,1), 'LineWidth', 1.2); xlabel('Time'); ylabel('y_1');
subplot(2,1,2);
bar([numel(t15) - 1, numel(t45) - 1]);
set(gca, 'XTickLabel', {'ode15s', 'ode45'}); ylabel('Steps taken');

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions